Hypertrophy is the process of growing tissues. In exercise physiology it normally refers to muscle hypertrophy or the increase of muscle mass. Muscle hypertrophy as a result of training is caused by an increase of cross section of themuscle fibers which form the muscle. This increase is caused by an increase in net protein production.

Muscle tissue is continuously subjected to remodelling, which means that there is a continuous production and breakdown of muscle tissue. This balance between production and breakdown is dependent on training and use of the tissue, genetic profile of the individual, rest and nutrition. There are several ways to realize muscle hypertrophy at a cellular level. Stimulating muscle protein production, or inhibiting muscle protein breakdown.

When protein production is greater than protein breakdown, hypertrophy will occur, which will lead to an increase in contractile proteins, the number of myofibrills within the muscle fiber, resulting in an increase in strength.

References:
-Baechle, T.R., Earle, W.R. (2008). Adaptations to Anaerobic Training Programs. Essentials of Strength Training and Conditioning. Third Edition. USA. Human Kinetics.