When you live in or worse grow up in a high crime neighborhood there is more social pressure to become a criminal. You live in an environment where that sort of thing seems more normal. You may think "our neighborhood is full of crime, so turning to a life of crime was just inevitable" and if you're the type of character who likes crime in the first place you'll feel more safe to get away with it.
By offering tax credits to people who have clean records to live in high-crime neighborhoods then good, law-abiding people move to high-crime neighborhoods and change the culture, the crime rate reduces, and so the social pressure that leads people, especially youth into a life of crime would decrease. This would mean less crime overall.
This would also help people socioeconomically, since over time crime would no longer be as associated as it is with certain neighborhoods. Home values would go up. Also the stigma of being in a bad neighborhood would decrease which would increase the confidence of youth in doing well in school and going to college, which in turn means more legitimate opportunities and would also be another factor in reducing the crime rate.
First of all, the main reason for the existence of high-crime neighborhoods is caused by harsh socioeconomic situations. In order to cure that sickness that plagues our democracies one must lower as much as possible poverty and extreme poverty. There's no magical solution and this problem will never diaper.
Now, imagine you are living in one of this poor neighborhood and you see rich people (or middle class) moving in and driving houses prices up. To add to the insult they get money for doing so (in an indirect form of course)! You are struggling daily to make sure your children won't have an empty stomach and you see someone who doesn't need more money getting more. That money should help them (at least this want I believe they will think).
You can't just change the ''culture'' without taking on the real problem (poverty).
There are many solutions some better than others but then it would require me forever to write them down. I'm glad you proposed something new.