How does the Arena work?

So, while yes the game advertises it as a "live" system, and you are technically playing against other players, you aren't playing them in real time. You are playing what they call "ghost data", and you aren't directly competing with other players, you're simply playing to see who gets the highest score against "bots" using other players previous game data.

Source (Emphasis mine):

Arena (known as Battlez until 7.5.1) is a player vs. player game mode in Plants vs. Zombies 2 that was introduced in the 6.6.1 update. This mode is unlocked after getting the Map of Space-Time-Ness from Ancient Egypt - Day 1 and obtaining at least six plants.

However, in the same article:

It has been noted that this mode does appear to be lacking a "real" matchmaking feature that pairs you with another player fairly. There have been reports of people that had low-leveled plants to still be paired with a person with max-leveled plants or still high-leveled plants. Players can also be paired against bots that solely calculate their score based on the players.

Some on reddit explains this a bit better:

[...] you can stay in the seed-selection screen as long as you could, and you could fast forward the game, and the fact you could exit to home screen and re-enter or even pause the game [...] So what you're seeing your opponent is what your opponent had done, albeit not instantly but maybe a few minutes or hours ago. You're literally playing against a ghost data of someone else, hence why you're actually playing "alone" during Battlez.