I can try to shed some light on this as I am currently reading Hellenika by Xenophon. The version is The Landmark Edited by Robert B. Strassler. It has an Appendix K that discusses specifically trireme warfare.

Triremes we're crewed by about 200 men. 170 rowers arranged in three banks, 16 sailors and officers to man sails, 10 armed marines, and 4 archers. Triremes we're built for speed, with long streamlined shape. They were powered primarily by the rowers, however sails were used to gain strategic advantages. The primary weapon was a bronze ram used to rupture opposing ships hulls. In preparation for battle ships would be stripped of spare equipment (sails masts etc) to improve speed. Rowers we're cramped at their station and were not able to really stretch out or lie down. Additionally there was not much room to carry much in the way of food. What this means is triremes generally would hug coasts in order to be able to put in in order to sleep, or to forage for food (or purchase food from local cities). In order to enable the crew to procure food, sailors including rowers who could be slaves, lower class citizens, resident aliens, or mercenaries all had to be paid or else desertion or defection were possible.

After a ship was rammed it wouldn't necessarily sink immediately but would instead be swamped and immobile. There wouldn't be any real obstacle preventing sailors from jumping overboard, or else clinging to the hull. After the battle was won, the victor would be able to rescue/capture floating hulls and men. Those ships could be repaired and potentially used again or else used to erect a trophy at the sight of the battle. Of particular value was the bronze ram. I think many Greeks may have been decent swimmers, but with the hull of the boat floating it may not have been necessary to know how to swim well.