There are several potential advantages:
Literacy: Being functionally literate is practically a requirement for modern life, and the greater your comfort with the written word, the easier it is to acquire knowledge. Reading to your child encourages them to think of books as "normal" things, and starts this process early.
Entertainment: One of the key things you want with a baby is the ability to self-sooth. Baby books are both intensely tactile objects, and look like the books that adults read, both of which are very attractive to babies.
Bonding: By reading to your child, you're committing to sitting and talking with them in a way you don't necessarily do the rest of the day. Unlike "conversing" with your child, where they obviously can't respond (which can get irrationally discouraging after 6-8 months), reading is a way to communicate that doesn't suffer from a lack of response.
Language: In the same way as singing, baby books are highly repetitive in terms of the words used, which helps to introduce them to basic words.
As far as when to start, we started reading from about 1-2 weeks, while walking the baby to sleep. At that point, it was mostly just so she could hear the sound of our voices. She's now a year old, and while she obviously can't read, her cardboard books are some of her favourite toys (and I generally hand her one of my books when I'm trying to change her and she won't sit still).