The exact details aren't super clear; somehow or another Bassett discovered Paul's riding habit and his tendency to shout out names about the races. Probably intrigued a bit, and part just humoring the kid in what might have been a passing fancy or fascination, Bassett lent him money for his first bet. Paul tells his uncle, "he lent me my first five shillings, which I lost," and then he used ten shillings his uncle gave him to win his first bet. From that point on, the bets most likely increased, and Paul was rather successful, because by the time his uncle figures out what is going on, Paul has over "three hundred pounds" on hand that Bassett keeps for him. Paul and Bassett formed a partnership of sorts--this is beneficial to both of them, beause Bassett gets the names of winning horses, and Paul has an adult supervisor who can place bets for him and keep his money and holdings safe. It works out well for both of them.
It's an interesting situation that eventually Paul's uncle wants to get in on. Bassett is an honest man when it comes to dealing with Paul's money, and they end up making quite a bit of it. I hope that helped a bit; good luck!