Rashi says that the source for eid echad being trusted for issurim is the pasuk that speaks of an individual shechting a korban, and all the kohanim eating from the shechita. Since we don't find that the pasuk requires there to be witnesses watching the shechita, we see that the shochet himself who is the eid echad is believed. Tosafos in Gittin 2b writes that the source of an eid echad being believed is the pasuk of וספרה לה by a woman who is a zava - she checks herself and her husband trusts her.
Rashi in Yevamos 88a writes that if an individual says that he knows that a piece of meat is שומן and not חלב he is trusted - דאי לאו הכי, אין לך אדם אוכל משל חברו, ואיןלך אדם סומך על בני ביתו. If we didn't allow the concept of eid echad, no person would be able to eat in their own home. This Rashi is usually quoted to be arguing on Tosafos and holding that the source of eid echad is merely a sevara, since otherwise it would be impossible to function normally. However, in light of Rashi in chulin who also cites a pasuk in the context of shechita, it seems that rashi in Yevamos isn't claiming to cite a source, rather proving that it must be true that we rely on an eid echad.
There is a very interesting Hagahos Ashri (Rosh 14) who writes that an eid echad isn't believed when there is a chezkas issur, but this is only if he is the class of people who are אנשים ריקים ופוחזים. Meaning, the same level of witnesses who the Torah believes if they are two, which includes even empty people who aren't at all learned, are not believed as individuals when there is a chezkas issur. But one who is kasher is believed even against a chazaka. According to the hagahos ashri, in a situation of chezkas issur, there are two classes of Jews, the lower of which isn't trusted unless they testify as a group of 2 witnesses.