Albert who has been told the month but not the day says he does not know the full date, he is not able to deduce it alone, in fact at this stage the only way he could deduce the date would be if there were a month with only one of Cheryl's possible dates in it and that were the month that he had been told. There is not.
He says he knows, that Bernard does not know the full date.
Bernard could know or rather deduce the full date if, say, the day he had been given were the 18th. The 18th and 19th are the only days that appear in only one month, June and May respectively. Since Albert says definitively that Bernard can not know the full date, or rather there is no possible day that Bernard could have been given that would enable him deduce the full date, we know that the month can not be May or June.

we can eliminate all the dates in May or June.

Then remain July 14, July 16, August 14, August 15, August 17.

Bernard did not know the full date, now he does. unlike me and you, he does know the day.

He knows what possible dates are available and if he had been told, say the the day is the 17th he would immediately see that there is only one possible date.
Only if the number he had been told were 14 then would he still not know the full date. Since he does now know the full date we can eliminate July 14 and August 14.

Then remain July 16, August 15, August 17.

Albert declares that he now knows the full date.

He would not be able to do this if the month he had been told were August as there are two dates in August.
So I can delete August dates and remains: July 16