It's not that hard to explain reversion to the mean. The schools in the lowest socioeconomic areas are generally, a priori, the worst of the bunch.
Consequently, just about anything will perform better. This is especially true if there is any ability to self-select (lottery is insufficient evidence as expelling "bad" students back to the public school is the same as self-selection).
When charter schools are compared in the suburbs, they generally perform worse than a public school.