Now I think there are 2 separate things, both very important. And there is a very strong link between these 2:
1.it is about the sound/silence thing.
2.it is about the use of iron tools.
In fact there is a point in asking about silence and user20490 is not the first one to do so. In Jewish exegesis there is quite an important discussion around this. Some would go as far as to say that the stones were carved somewhere else:
Mishneh Torah, The Chosen Temple 1:8: The Temple is to be built
with large stones or bricks if large stones are not available. The
stones are not carved on site, rather they are carved and dressed away
from the Temple Mount and brought in afterwards, as it says "Large,
expensive stones for the foundation of the House, hewn stones" and it
[also] says "the sound of the axe, nor any other metal tool was heard
in the Temple while it was being built."
This idea is used quite often in various Christian commentaries too. See one example from many others:
The stones for the temple were prepared at some distance from the
construction site. This may be for practical rather than religious
reasons, as seen in Assyrian reliefs depicting completed stones and
statues being removed from a quarry. The erection of the temple went much faster by utilizing pre-cut and pre-fitted materials moved on rollers to the temple site. [J. H. Walton, Zondervan Illustrated Bible Backgrounds Commentary (Old Testament) Volume 3: 1 & 2 Kings, 1 & 2 Chronicles, Ezra, Nehemiah, Esther, Grand Rapids, MI: Zondervan, 2009, p. 30]
However, the stress of these is, I would say, on the fact that all of the fuss was necessary in order to make sure that no sound of any metal was heard in the Temple. No sound in the Temple seems to be a very important matter. Why is that no sound? I found another old comment on this:
Mekhilta d'Rabbi Yishmael 20:22:2 (I Kings 6:7) "And hammers,
chisels, or any iron tools were not heard in the Temple when it was
being built"? In the Temple they were not heard, but outside (where
they were hewn) they were heard. (Exodus, Ibid.) "For if you lift your
sword upon it": R. Shimon b. Elazar was wont to say (Devarim 27:6) "Of
whole (shleimoth) stones shall you build the altar of the L rd" —
stones which repose peace ("shalom"). Now does this not follow a
fortiori, viz.: If the stones of the altar, which do not see or hear
or speak — because they repose peace between Israel and their Father
in heaven, the Holy One Blessed be He says: Do not lift iron upon
them, then one who reposes peace between a man and his wife, between
one city and another, between one nation and another, between one
government and another, between one family and another — how much more
so will he not meet with adversity!
It does make sense, isn't it? After all, in 1-2 Kings God is more like "a soft whisper":
1 Kings 19:11-12 The Lord said, “Go out and stand on the mountain before the Lord. Look, the Lord is ready to pass by.”A very powerful
wind went before the Lord, digging into the mountain and causing
landslides, but the Lord was not in the wind. After the windstorm
there was an earthquake, but the Lord was not in the earthquake. After
the earthquake, there was a fire, but the Lord was not in the fire.
After the fire, there was a soft whisper [קֹול דְּמָמָה דַקָּה׃].
Silence is a sign of the (future)presence of the Lord.
2. The use of iron tools
First of all, it seems that this was a more general ANE taboo on this respect. See this:
An early Sumerian account of Gudea’s building a temple for his god
insisted that there be no noise around the area of the temple during
the building project. Masons used large picks (weighing thirty or
thirty-five pounds) for quarrying and smaller picks (weighing twelve
or fifteen pounds) for the shaping of the stone. Those found in
excavations have iron heads of various shapes with wooden handles.
Iron-headed sledge hammers and long double-handled saws were also
portrayed on Assyrian reliefs. [V. H.Matthews,
M. W. Chavalas & J. H. Walton, The IVP Bible background commentary :
Old Testament (electronic ed.) (1 Ki 6:7-35). Downers Grove, IL:
InterVarsity Press., 2000]
Last but not least, I think the full answer should be found in Genesis 4.
After killing Abel, Cain is going in the land of Nod, east of Eden, where he has a son, Enoch. Then, Cain is building a city right there, east of Eden. Now, from Genesis 2:8 we know that it is in the same place, east of Eden, that God planted an orchard and there he placed the man he had formed. There is a paralel here, I think: on one hand there is the orchard made by God and on the other there is Cain's city. Let's return to Genesis 4: one of Cain's descendants is Lamech, who took 2 wifes. The second of Lamech's wifes, gave birth to someone called Tubal-Cain:
Genesis 4:22 Now Zillah also gave birth to Tubal-Cain, who heated metal and shaped all kinds of tools made of bronze and iron. (NET)
Therefore, from a Biblical chronology, it is only with Tubal-Cain that we have metal, bronze, iron. And as the narrative goes on, we learn that this Lamech, father of Tubal-Cain, was not really a gentle kind of person. (see Genesis 4:23-24). He was not peaceful at all.
Then, in Genesis 4:25-26, there is what I would call an antithesis of all that we have seen before:
Genesis 4:25-26: (25) And Adam had marital relations with his wife again, and she gave birth to a son. She named him Seth, saying,
“God has given me another child in place of Abel because Cain killed
him.” (26) And a son was also born to Seth, whom he named Enosh.
At that time people began to worship the Lord. (NET)
That was so brief. Adam > Seth > Enosh > worship the Lord. No buildings, no tools, no metal, no bronze, no iron, no killings, no revenge, no fear, no fury. Just simply, plainly: worship the Lord.
A possible conclusion + answer: David did not manage to build up the Temple, because he was a man of war. Solomon is a man of peace. If you look at Genesis 4:1-14, there is some of "David's style". Whereas if you look at Genesis 4:25-26, there is some of "Solomon's style". I do think there is a paralel between the building of the Temple and these.
And to conclude:
silence is a sign of the (future) presence of God
no iron tools is about: underlining the link with Enosh and highlighting the distinction from the lineage of Cain. This should be available for Exodus 20:25 (see also Deut 27:5, Joshua 8:31) too. Remember, Cain is the one who built a fortress east of Eden and started to use iron tools, instead of seeking peace with the Lord, in the orchard east of Eden, where no iron tool was necessary. The Garden of Eden and the Temple are linked in the Bible by many shared motifs and by shared imagery. Do find a helpful source for this at the end of this link, please.