Some people pray quietly (except where praying aloud is recommended or required, as for the first verse of "Sh'ma"), others aloud, by which I mean loud enough to be heard from a few feet away, and yet others quietly but with occasional phrases said aloud. But at n'filas apayim virtually everyone says the whole thing quietly: why?
I don't have a source offhand, but I remember learning that it is connected to the idea that we don't publicly announce our sins between man and G-d, and it is considered Chutzpah if we do publicly announce our sins that we sinned against G-d. (Rambam Hilchot Teshuva 2:5 - english).
Since we are confessing our sins, we do so quietly.
Taamei HaMinhagim, entry 133 quotes the Mateh Moshe to explain why we cover our faces during Tachanun. When the Jews would prostrate themselves in the Beit HaMikdash, there was (miraculously) a space of 4 cubits between every person. This was in order that one person would not overhear the confessions of another person, which would be embarrassing.
We therefore similarly hide our faces, as if we don't see or know what our fellow congregants are saying. [In a footnote, he connects this to a Shaarei Teshuva, who says that one should fall on their faces when the congregation is doing Nefilat Apayim, even if he is not at that point in the prayer].
Although he does not say anything about praying quietly, maybe this is the reason why people do Nefilat Apayim quietly, since we don't want that prayer to be overheard.
The Torah tells us (Bereshit 46:29 and Rashi there) that when Yosef reunited with Yaakov, he fell on Yaakov's neck and wept. Yaakov however was silent. Rashi tells us it is because he was saying Shema.
I just came across this Shaar HaKollel(Vidui V'Nefilat Apayim 6). There, he quotes the Yalkut Chadash quoting the Asarah Ma'amarot, saying that we learn Nefilat Apayim from Yaakov's reunion with Yosef. Yaakov was saying Shema, and if he said Shema he definitely followed up with Shmone Esreh, in order to juxtapose Geula and Tefilla. Yaakov then continues (Bereshit 46:30), 'Now let me die, since I have seen thy face, that thou art yet alive.' This statement, the Asarah Ma'amarot says, is the self sacrifice of Nefilat Apayim.
If so, perhaps we can say that the reason people say nefilat apayim quietly is because Yaakov said Shema quietly when he reunited with Yosef.
My own thoughts.
Similar to Shemona Esrei, Tachanun is an oppurtune time for a person to pray for personal needs. I would imagine that since people are praying for personal salvations it is done quietly.