Question

While it was illegal to teach slaves to read, how did some slaves end up learning to do so?

Answer 1

Some learned from other slaves and masters. While some countries wanted then to be able to read and write. 

Answer 2

Some slaves learned how to read through other slaves, or in some cases the masters of slaves teach them how to read.

For example, Frederick Douglass learned how to read from his mistress, Miss Auld, who taught him simple letters and words.