An arena is named after what covered its floor.
The Latin word harena meant "sand" or "a sandy place." In the Roman amphitheater, the central fighting space was strewn with sand. Wikipedia says the Colosseum's wooden floor was covered with sand. Etymonline says the Latin word later referred to the enclosed central space itself.
English borrowed the word in the 1620s, according to Etymonline, with the sense "place of combat." The English spelling dropped the Latin initial h. Etymonline lists related forms in Spanish, Italian and French.
The broader meaning came later. By 1814, Etymonline records, arena could mean the "scene of contest of any kind." That is the sense behind the political arena, the arena of public debate, and the modern sports arena, which has no sand on it at all.
Where did the Romans get harena in the first place? Nobody knows for certain. Wiktionary gives two accounts. One says it comes from an earlier form *hasena, which would match a Sabine word fasena, and that the word is possibly Etruscan. It cites an Etruscan word faśena, meaning sand, ashes or human remains, as a parallel. The other, from the Latin scholar Michiel de Vaan, suggests a verb with a particular suffix. Wiktionary notes that no candidate verb fits, so the origin stays unknown. Etymonline is similarly cautious and says Etruscan is a possibility.
The sand left one more word in English. Arenaceous, first recorded in the 1640s, means "sandy" and comes straight from the same Latin source. By 1870 it had also picked up a figurative sense of "dry."
So when a stadium announcer talks about stepping into the arena, the word is still doing what it did in Rome. It points at the floor.