Okay. I'll call the times T1, T2, T3. As in:
At T1: Albert says that doesn't know when Cheryl's birthday is, but he knows Bernard doesn't know either.
At T2: Bernard agrees that he didn't know at T1 - but now, having heard what Albert said at T1, he says he knows.
At T3: Having heard what Bernard said at T2, now Albert knows too.
I'm not confident I've made it sound any less confusing, but I know what I mean, which I suppose is a start of sorts. Anyway: At T1 Albert can be sure it isn't May 19 or June 18. If it were, Bernard would know (because there is only one 19 and only one 18 on the list). The only way Albert can be sure of that is if he wasn't told May or June
at all.
So now, at T2, Bernard can tell that Albert was told July or August. If he can use that information to eliminate all but one possibility, he must have been told the number "16".
So at T3, having discovered that Bernard was able to get it purely from that, Albert knows that Bernard was told the number 16. And remember that Albert already knows the month.
I don't know if that makes it clearer, but basically that's why A_Seagull is right. The puzzle doesn't explicitly say that Albert and Bernard are having a conversation, which I think is what threw Badger off. What I haven't figured out yet is why anybody would want to be friends with a pain in the ass like Cheryl.