Jesus just got through saying, "I was sent only to the lost sheep of the house of Israel" in the previous passage. But like the Canaanite woman who wouldn't be turned away, now the multitudes come running after him, and we know this crowd was largely Gentile because Matthew says they "glorified the God of Israel." Didn't they receive the memo that Jesus was only sent to the house of Israel? Haven't they heard that it's not right to feed the children's bread to dogs? Even if they have, do they care?
And Jesus obliges them. What might he be thinking at this point, seeing these desperate Gentile crowds? Well, fresh in his memory is that astounding answer given by the Canaanite woman, that even a dog like herself would happily feed on the crumbs that fall from the master's table. Her faith, humility and insight amazed Jesus, and he could not turn down her request for her daughter's healing. Prior to that encounter Jesus had had a run in with the Pharisees, who condemned the disciples for not following the tradition of washing their hands before they ate, who then became offended when Jesus pointed out how they transgress God's law to uphold the man-made tradition of corban.
So Jesus was sent to the lost sheep of the house of Israel, but the leaders of Israel are already hardening their hearts, violating God's commands in favor of the teachings of men. Meanwhile, the Gentile woman had more faith in her little pinkie than all those Pharisees put together . . . Jesus had to be thinking that the Canaanite woman is more of an Israelite than the ones who are born into the privilege. And surely these Gentile multitudes clamoring for his attention are also looking for some crumbs for themselves.
So he heals them, their lame and crippled and blind and mute. The Pharisees were too righteous and whole to have need for the Messiah. But the ones who feel so unworthy they ask only for crumbs will find themselves seated at the banqueting table with Abraham, Isaac and Jacob.