Is that because the original texts are indented too or something, so that it looks like a poem?
No. The old manuscripts (at least as far as I am aware) do not indent poetry. However, since there are so many, I would advise looking up one of the projects that is digitizing the manuscripts so you can see them for yourself.
Edit: you can view a number of manuscripts online at this page (http://www.csntm.org/ ). There are a lot of manuscripts here, though I did note one just now that may be indenting, though without being able to take the time to translate, i can't tell if the passage in the image is poetic. Source: http://www.csntm.org/Manuscript/View/Rahlfs_964
Or is it fully based on the early translator's decision?
A) At some points, it is an issue of interpretation. For example, some Christians that hold to Exclusive Psalmody (the view that only the book of Psalms should be sung during corporate worship) deny that Philippians 2:5-11 is poetic in nature, and so would like to see it treated as straight exposition.
Edit: Here is a link to Brian Schwertley's defense of Exclusive Psalmody, in which he argues against against the idea that there are hymn fragments in the NT http://www.reformedonline.com/uploads/1/5/0/3/15030584/exclusive_psalmody.pdf
Dr. Gordon Fee also wrote a piece questioning whether Philippians 2:5-11 is a hymn fragment, though I don't know if he is an Exclusive Psalmodist. http://www.biblicalstudies.org.uk/pdf/bbr/philippians_fee.pdf
B) In other cases, it's an issue of where recognizing that the biblical author is quoting from a poetic source. If you're quoting from the Psalms, for example, then you're certainly quoting poetry.
C) Another angle to consider is that just like we have certain poetic styles in English (and other languages), so does the Hebrew and Greek. While my Hebrew is quite rusty, I seem to recall that Hebrew poetry has a slightly different vocabulary and grammatical structure as compared to non-poetic Hebrew. When a translation committee sees these signs, they in turn give it a special indentation in the translated work to mark it as poetry.
Edit: citations of specific grammatical differences in Hebrew: Taken from "A Grammar of Biblical Hebrew" by C.L. Seow ISBN 0-687-15786-2 --
- The absence of the article in poetic Hebrew is, in fact, quite typical. (page 88)
- Excursus E covers poetic Hebrew (pages 157-159) and its idiosyncracies: one example -- "In Hebrew poetry, a single preposition in one line may also govern a noun in the parallel line." (page 159)