Thrust and angle are more basic, simple relationship. Take that as the baseline. Excess thrust yields climb angle.
Now, remember that power is thrust times speed.
Then, take thrust and angle, and multiply them both times speed. Thrust becomes power (from the above law). And angle becomes rate. Not really exactly a multiplication, but... a certain steepness of angle, with a higher speed than before, gives a higher climb rate than before. It’s proportional.