In running, things are bit more complicated because it's a weight bearing sport which means that each race takes a bigger toll on the body mechanics.
At 20 years old, I decided to try an Ironman. Coming from a cycling background, I jump started my running by racing each weekend. In one month, four weeks after having started running (EVER), I ran a road marathon, a 45k ultra, a 25k race and a half marathon. You can imagine the outcome of that "stunt".
Hence, when I say "race lots" I'm thinking of shorter distances, mostly anything up to 10k. I keep the number of 10 milers and half marathons low. Races provide a training stimulus that is hard to train. It's easier to push boundaries in a race than on a track in a solo workout. Plus, forced pace changes give you that little bit of an extra challenge that can't be simulated.
As for me, I really hope to get a grip on my racing this year. I just know that I won't be nearly as fast in my A race if I don't do tons of "C" races. I say "hope" because my schedule this year is very, very different from what it was only five years ago. I'll be all over the place. But I look at it as a great opportunity to jump into local races around the world.