Henry said:
I'm guessing they don't, and that's the secret.

If the group is in tune with what they're doing, the DM and players can probably guesstimate what is going on, op-attacks, etc. with the players trusting the DM as to adjudicating who is in range to get hit by what, etc.
Pretty much. Playing with minis gets you used to the idea that everything is clearly defined and there's an objective, rules-based answer to every question, which can be resolved without recourse to the DM. Playing without minis requires one to break out of that mold.
Non-battlemat combat is much more loose and freewheeling, and involves a lot of on-the-fly judgement calls from the DM. Instead of the wizard carefully positioning her
fireball on the grid, she asks, "How many can I hit with a
fireball?" And the DM replies, "Three of them, or four if you're willing to hit the paladin as well." Then the wizard decides whether to fry three enemies, or four plus the paladin, or do something else, and the game moves on.
Now, when this happens, the DM is not consulting an elaborate mental map. He's just tossing off a number that "feels" right.
Opportunity attacks involve similar handwaving. Typically, a player will say, "I move up and attack the orc chieftain," and the DM will reply, "That's going to provoke two opportunity attacks from the grunts." Then the player can either say, "Okay, I'm doing it anyway," or, "Oh, then I'll attack one of the grunts instead."
It seems like it should be slower than miniatures-based combat, since you're asking the DM questions all the time, but my experience is that it actually goes faster. The time spent asking the DM stuff is more than compensated for by the time
not spent counting off squares, figuring out exactly where to place spell effects, and plotting just how to move so as not to provoke AoOs.
As others have said, however, this style of play requires a lot of trust between players and the DM, because
everything boils down to DM judgement calls. The players have to trust the DM to make those calls fairly. The DM has to trust that when a player says, "What? I didn't realize they were positioned like
that... if I'd known, I wouldn't have done that," the player is telling the truth and not trying to fudge an advantage. If you have that trust, it's a fast-paced, immersive, and exciting way to play. If you don't, it will degenerate into a hellacious argument.