When you avouch, you assert that something is true. An accused man might be acquitted if several eyewitnesses avouch that he was nowhere near the scene of the crime.
Avouch is an extremely old-fashioned way to say another (only slightly old-fashioned) word, avow. Use either word to mean "assert" or "declare." While it's unusual to hear people use avouch, it's common to say something like, "I'll vouch for my friend," meaning you'll give evidence or make a guarantee on your friend's behalf. The root of avouch is the Latin advocare, "call to as a witness."