One advantage to playing on your home court is being used to the stadium. Players play more in their own court than in any other team's court. This means they're used to the playing-surface and likely feel comfortable on their own court.
Playing on your home court means playing in front of your own fans. Hearing beloved fans encourage you when you make a shot goes a long way for a team. It adds to the comfort level and affects the opposing players too. The fans can get in the heads of the away team and rattle the players, throwing them off their game. It's hard to make a clutch free throw when the whole stadium is yelling at you!
The simple answer is yes. In the NBA, home teams win about 60% of their games. In the playoffs, home teams win approximately 65% of their games.
In the NCAA, it is believed that home court advantage is much more important for some teams than others. Overall, home court advantage appears to be even more prevalent than in the NBA. In the 2017-18 NCAA season, home teams won approximately 68% of their basketball games.
Home field advantage clearly does give teams an advantage, since home teams win the majority of games. However, it's not a guaranteed win.