There are many answers here but most of them are based on pure speculation and fanciful theories with no evidence to back them. So I decided to post a scholarly based answer here. I hope this will be of help to anyone researching this fascinating topic.
Conspiracy theory is lacking evidence
The scholarly consensus is that the three recensions of the antediluvian and postdiluvian chronology of the patriarchs (SP, MT, LXX) were not doctored by some Rabbis or any other group of scribes after the death of Christ, contrary to what the OP implies in his question. That is pure conspiracy theory and highly implausible (HB Smith is one of many who propagate this myth), no group of scribes that late would've possibly been able to get away with such big scale revision of the biblical text. Instead these revisions or recensions date back to way before the Septuagint was translated (3rd century BCE) or the MT standardized. Scholars (So Klein, Hendel, Tov) properly refer to these recensions as proto-MT proto-LXX and so forth, because they evolved way before these texts morphed into their current form. The Rabbis or the Masoretes of the MT did not change the text or anything, but they received these chronologies in their current corrupted MT form, lock stock and barrel. And the same goes for the LXX translators, they didn't change anything but were working with a text that has already been corrupted centuries before.
The most plausible and convincing theory I've seen so far is the one advanced by Ralph Klein, which anyone can read here. I will summarize it shortly, and I will focus primarily on the antediluvian chronology.
But first it is important to note that with the exception of Lamech, the LXX and MT both agree on the total lifespan of the patriarchs from Adam-Lamech, but disagree in regards to the age of fathering, the LXX usually adds 100 years to the age of fathering (which subsequently pushes off the onset of the flood by a 100 years) and subtracts 100 years from the remainder, so the total lifespan is retained both by the MT and the LXX. The SP however has a shortened lifespan for Jared Methushelah and Lamech (the three problematic patriarchs), as we will discuss below.
Klein's theory: Evidence from the SP
Now in regards to the lifespan it is quite easy to ascertain what the original text (which Klein calls the archetype) had. According to the SP the flood occurs at the year 1307; in a bizarre way all three aforementioned patriarchs die in exactly that year according to the SP (which by itself seems very suspicious), a peculiarity not found in the LXX or the MT, it seems obvious that the text in the SP has been tampered with and doctored to resolve a flood related issue, and that the LXX and MT are more authentic, at least in regards to the lifespans of the patriarchs which they both seem to agree on (with the exception of Lamech; see end of post).
The Notorious Flood Problem
Klein proposes that in the archetype Jared Methushelah and Lamech all survived the flood, this posed an obvious problem as it seemed inconsistent with the flood narrative which implies that no one except for Noah and his children survived the flood, so this was dealt with in different ways. The proto-SP as we have already seen did straightforward doctoring, they shortened the lifespan of the three problematic patriarchs and eliminated the problem altogether. The proto-MT and proto-LXX however dealt with this differently. Instead of shortening the lifespan, they just moved up the age of fathering of certain patriarchs, this delayed the onset of the flood by a few hundred years and at the same time retained the lifespan as recorded in the archetype. However the proto-LXX and proto-MT don't agree completely, they each put a slightly different spin on it.
The Archetype: Reconstructing the original text
According to Klein, in the archetype the flood occurred in 1342 (closest to the SP value), but Jared Methushelah and Lamech's lifespans were too long as they survived long after 1342 (see list below). To resolve this inconsistency the proto-MT added 100 years only to Jared Methushelah and Lamech, the three problematic patriarchs (in the case of Methushelah they added 120 years. Indeed he dies in the year 1656, the year the flood occurs according to the MT's chronology), the rest were left as they were in the original text, since they posed no problem at all. The proto-LXX however, perhaps to ensure consistency added 100 years to all the patriarchs; while this takes care of Jared and Lamech, the problem persists with Methushelah as he survives 14 years after the flood. This remains a problem for Klein's hypothesis. Klein speculates that the proto-LXX scribes were not comfortable with adding tens of years (as the proto-MT added: 120), only hundreds. Thus the Methushelah problem remained unresolved in the text of the LXX. Klein himself notes that this explanation is not so satisfactory, and leaves the question open as to why the proto-LXX added 100 years to each of the patriarchs.
This is how Klein reconstructs the hypothetical age of fathering (f) for the three problematic patriarchs in the archetype:
1. Jared: f. 62 (A.M. 522.) Total lifespan 962. (522-1422)
2. Methushelah: f. 67 (A.M. 654) Total lifespan 969. (654-1556)
3. Lamech: f. 88 (A.M. 742) Total lifespan 753. (742-1407)
Flood (A.M. 1342)
(To find the time of the flood always add 600 to the corresponding time period (A.M.) of f of Lamech. In this case it's 742 + 600 = 1342)
As anyone can see, the proto-LXX just added 100 years to f of all the three patriarchs, and the proto-MT added 100 to f for Jared and Lamech, and 120 for Methuselah, as 100 wasn't enough to resolve the flood problem (Indeed the flood inconsistency remains in the LXX). Proto-SP kept the original value of f for Jared and Methushelah, but not for Lamech, this was probably due to scribal error, and not intentional (see Klein p. 261-2).
That the motive of those who doctored the proto-SP text was to synthesize the lifespan of the patriarchs with the flood narrative is clearly evident from the fact that the SP follows the MT and the LXX for all the lifespans of the patriarchs from Adam-Lamech except for Jared Methushelah and Lamech which all end coincidentally at 1307 the year of the flood. This clearly betrays the motive for the three recensions. While this motive is not readily apparent in the MT and LXX, it is safe to assume that they all share a similar motive. Especially since the MT also somewhat gives this away in the lifespan of Methushelah which coincidentally ends the same year of the flood, 1656 (this was achieved by adding 120 years to the age of fathering for Methushelah as we discussed above). There's no need for further speculation as to why the proto-LXX and proto-MT added hundreds of years to the then-existing chronology, or why the SP shortened the lifespans of certain patriarchs, when we study the SP and MT and LXX together and compare them, the motive is clearly revealed: to synthesize as much as possible the chronologies in Gen. 5 with the flood narrative.
The Postdiluvian Chronology in Gen. 11
Unfortunately, the motive for the chronological recensions in gen. 11 (in the LXX, SP and MT) is not as easy to discern. The LXX and SP both add 100 to almost all of the patriarchs and 50 to Nahor. Klein favors the MT in almost all instances, and ascribes most of the LXX variant dates for f to scribal errors, but I remain unconvinced. While Klein's hypothesis for the antediluvian patriarchs is well supported from the texts, evidence is lacking for the postdiluvian patriarchs.
@Hannes' theory - that the proto-MT deflated the years for f so that Abraham would be the only one fathering miraculously at 100 - at first seems plausible, but it doesn't seem to account for the deflation of Nahor in the MT (since the f for Nahor in LXX and SP are both 79, well below 100), or why the original f value for Shem was retained in the MT, since Shem's fathering at 100 technically poses a problem as well in this framework. Although similar problems were posed in our "flood problem theory" above (in the LXX the age of Methuselah remains a problem), there were other evidence to point this way, whereas over here no such evidence exist. Klein and Tov suggest a proto-LXX inflation theory for most of the postdiluvian fathers, and the motive, according to them, is that no more than four generations should be alive at the same time, since in the MT Shem and Eber are still alive during Jacob's lifetime, this may have been viewed as a problem by the proto-LXX and proto-SP scribes (The MT's chronology would obviously create tension in the text that states that Abraham died an "old man" and "full of years", as Shem and Eber both outlive him!), and this was achieved through inflating the value of f for most of the patriarchs by a 100 and 50 for Nahor. While this theory is surely likely, further evidence is lacking.
So to summarize, the postdiluvian chronology seems to be a double edged sword, if we add years to f consistently (as we find in the LXX) to mitigate the problem of Abraham dying prematurely we create a different problem, namely the problem of Abraham not being the only patriarch fathering at 100. Similarly, if we deduct years from f consistently to mitigate that factor, then Abraham dies prematurely as Shem and Eber outlive him. Both of these scenarios are not favorable. It's hard to determine what was the original chronology, and which of these inconsistencies were deemed more problematic to the proto-MT and proto-LXX scribes enough to warrant a uniform emendation of the text.
It should be noted, that unlike gen. 5 where LXX and MT both agree on the total lifespan, in gen. 11 LXX's total lifespan is almost always 100 years more than the MT, and in some cases even more. The reason for that is simple, because unlike the antediluvian chronology in gen. 5, the postdiluvian chronology in gen. 11 does not summarize the total amount of years for each patriarch, thus the proto-LXX does not feel obligated to subtract the 100 years it added to f from the remaining years of the patriarch as they had to do with the chronology in gen. 5 to retain the total amount of years which was clearly summarized in the text.
It should be noted that in the case of Lamech's age for f, which is 182 vs. 188, Klein sides with the LXX age of 188, and ascribes the 182 of the MT to a scribal error. In regards to the lifespan of Lamech, the only case where the LXX disagrees with the MT, Klein sides again with the LXX which has 753, and ascribes the 777 of the MT to a conflation between the Lamech of 4:24 with the patriarch Lamech listed in the chronology chapter 5. However this remains speculative.