The scientific word for muscle growth is hypertrophy. This basically means that the cells (muscle fibers) in your muscles and the stuff between the cells get bigger. (1). This is different from hyperplasia, which is an increase in the number of muscle fibers in a muscle.

You muscle fibers are packed with proteins called myofibrills, and during hypertrophy the cell makes more of these. Myofibrills are what make muscles contract, so more myofibrills = more strength. (2,3). This happens in response to lifting weights!
More myofibrills also means that muscle fibers need to grow cross-sectional area to make room for them, resulting in muscle growth (4). This is main type of muscle growth, also known as myofibrillar muscle protein synthesis.
Old theories spoke of something called sarcoplasmic hypertrophy which means muscle growth by growth of the watery part of muscle cells. This type of hypertrophy occurs parallel to regular hypertrophy and cannot be “isolated” with high rep light weight training as bro science puts it. Both low rep and high rep light weight training to failure result in myofibrillar hypertrophy and thus sarcoplasmic hypertrophy. (5)

1. Vierck et al. Satellite cell regulation following myotrauma caused by resistance exercise. Cell Biol Int 24: 263–272, 2000.

2. Paul, AC and Rosenthal, N. Different modes of hypertrophy in skeletal muscle fibers. J Cell Biol 18: 156: 751–760, 2002.

3. Tesch, PA and Larsson, L. Muscle hypertrophy in bodybuilders. Eur J Appl Physiol Occup Physiol 49: 301–306, 1982.

4. Toigo, M and Boutellier, U. New fundamental resistance exercise determinants of molecular and cellular muscle adaptations. Eur J Appl Physiol 97: 643–663, 2006.
5. Burd et al. Low-load high volume resistance exercise stimulates muscle protein synthesis more than high-load low volume resistance exercise in young men. PLoS ONE. 2010;5(8):e12033.