How do you gain strength without gaining big muscles?

Well, there is a balance between strength and muscle size.

If you want to get really strong, you'll have to accept having really big muscles.

But if you just want to get stronger, then the key is to exercise on a sensible level.

If you keep the weights so that you can do 3 sets of 10 repetitions, or even 4 sets of 10 reps, muscle growth won't be that much.