When do Scout and Jem start to trust Boo?

They begin to trust Boo when he starts reaching out to them. He does this by leaving a succession of objects for them to find in the knothole of a tree. This is Boo's sole connection to the outside world. The children realize this, and appreciate the opportunity to communicate with him in this unusual way.
They also feel somewhat privileged as they've now gained a glimpse into Boo's world, a world that had previously remained firmly closed. Like just about everyone else in Maycomb, Scout and Jem have tended to regarded Boo Radley as a kind of freak or boogie-man, a man of mystery whose whole public identity has been cobbled together out of scraps of local folklore, hearsay, and gossip. But now that Boo's started reaching out to them, they come to see a different side to him, and a firm bond of trust gradually begins to form.