Your body adapts to training and gets stronger/bigger/faster/smaller because of the neural, muscular, hormonal, and skeletal changes that are the result of chosen training stimulus.
Is it possible, then, to get stronger without getting bigger? Yes, it is. It all depends how strong one wants to be. I came across an article on the Internet a while ago.
The author described strength as a “skilled act.” I really liked this description, as it implies you can teach your body to be strong.
To start, let us focus on sports that require great amounts of strength and/or power and at the same time involve weight limitations – combat sports (i.e. boxing, MMA, BJJ), gymnastics, ballet (Yes, I do consider ballet dancers athletes), ice skating, and climbing.
While sports like boxing or MMA have explicit weight groups, in other sports like ice-skating, climbing, dance and gymnastics the lighter and stronger you are the better off you are.
Training all of the above, or even training women (“I don’t want to get bulky” is probably what personal trainers hear first from most female clients) should focus mostly on the neural adaptations to strength training.
Let me make it clear – research shows it is not possible to induce only neural or only muscular changes.
Both always take place, but there are certain protocols that allow increases in strength without or with minimum increase in the muscle cross section.
The initial strength gains when you start strength training come mostly from neuromuscular adaptations rather than hypertrophy.
If you notice your muscles “pump up” during or shortly after the session, don’t panic or be too happy (depending of what your goal is). This is a short term increase in size caused by fluid retention. It should fade away within 60-90 mins after the training.