This was initially part of Blackfriars Priory.
From St Etheldreda: From about 1590 onwards the gatehouse become a building where Catholics secretly worshipped and it was repeatedly raided by government priest-hunters. Purchased by Shakespeare in 1613, though there is doubt about whether he ever lived here. The St Etheldreda website has evidence about this property that suggests Shakespeare was, late in life, a closet Catholic. Shakespeare left the property to his daughter, Susanna.
The picture source tells the story of a building which was probably this gatehouse. In 1623, 7 years after Shakespeare's death, the attic floor collapsed during a Jesuit service which was attended by 2-300 people. 90 died.