The counter tracks whether or not the person getting hit was still in hitstun. After you get hit, there's a variable period where you the game won't pay attention to any button inputs, so you're really a sitting duck. If you get hit again during this period, the combo counter goes up. True combos are inescapable because they hit people who are still in hitstun and therefore have no escape options.
The reason combos don exits in Brawl is that hitstun is super short for most moves, to the point that it's relatively rare that the attacker can finish an attack animation and follow up before hitstun ends. If the opponent gets even one frame out of hitstun, a lot of characters (e.g. Marf, Bowser, Zamus, Squirtle) have moves which either attack on frame 1 and hit the attacker out of the string, or have invincibility frames on startup, which means pretty much the same thing.