Well, for the fun of answering it:
Let's start out with the vocabulary.
Costard was used by Shakespeare to mean head. This is supposed to be a comical term. According to the Shakespearean Lexicon it is a "ludicrous expression of the head."
Brast interestingly is not a Shakespearean term; however, it was used by Chaucer. The word brast means to burst, or in other words, to crack open. The word was originally burst and was believed to be corrupted by the Old Norse influences in England.
Now the term was also used in the 1530s to not only mean to physically burst but also to figuratively burst due to excitement. "I'm bursting with the news of the new baby." "You brast my costard with news of the new baby."
In the case of the Shakespeare Stealer, the phrase "brast my costard" is referring to the knock on the head Widge gets when he tries to listen at the door.
Wait nevermind I got the answer it means that he smashed Widge's head by slamming the door.