Exploitive play refers to strategies that may not generally be optimal, but which are designed to exploit previously observed flaws in a specific opponent's strategy. Simple example, you're playing a series of Rock Paper Scissors games. The game theoretic optimal, and unexploitable strategy for this game is to pick randomly every time 1/3 rock, 1/3 scissors, 1/3 paper. But if you do this you also won't be able to exploit flaws in any opponent's non-optimal strategies, so you are guaranteeing yourself of having no edge.
So what if after 10 trials your opponent has gone Rock every time? You might get the idea that on the 11th trial he's a bit more likely to pick Rock than anything else. So instead of selecting randomly, you should play Paper. If this guy never adjusts and keeps throwing Rock, you can keep throwing Paper. So your new strategy of "Paper every time" is an exploitive strategy against his "Rock every time" strategy. But by doing this you yourself also become exploitable, clearly "Paper every time" is just as bad a general strategy as "Rock every time". It just happens to exploit your opponent's particular weakness.
For super detailed discussion of game theory concepts as applied to poker see
Mathematics of Poker.