To the Romans, a theater was a semi-circular structure. Greek theaters had their stage area at the bottom of a hill. Seats were built in courses up the side of the hill. For best hearing, the whole would be curved around the stage. The Romans didn't need hillsides; they could build up seats in tiers, like an artificial hill. Still, the semi-circle was the most efficient form for a theater in which the spoken or sung word was the primary means of presentation. Like Shakespeare's Globe, most of the scene was set in words, not by means of set decoration.
If you put two theaters together along their back ends, you get a full circle, with tiers of seats all around. The prefix amphi- or ambi- shows up in other words, implying doubleness, like "amphisbaena," "ambivalent," or "ambidextrous." So an amphitheater is just a round stage or field with seats in circular tiers.
(A stadium was originally a footrace course, one stadion long. A stadion was about 200 yards. So, an amphiteater was used for public spectacles in the round, while a stadium was a track and field complex. A chariot-racing track was called a circus by the Romans.)