If Albert knows that Bernard doesn't know, the month can't be either May or June, since either of those contains a possibility of a unique date (May 19 or June 18). Therefore Cheryl's birthday is in July or August.
So Bernard knows it's in either July or August, and this is enough for him to determine when the birthday is. It can't be on 14th, because this wouldn't enable Bernard to choose between the two months. So we're left with July 16, Aug 15 or Aug 17.
Now back to Albert: he can deduce the answer given Bernard's previous answer. If the birthday were one of Aug 15 or Aug 17, Albert wouldn't be able to tell which. Given that he knows that Bernard knows the date and that this enables him to deduce the date, the only possible option is July 16.