Ultimately, muscle fibers hypertrophy by increasing their protein content. Strength training, believe it or not, increases the protein content of your muscles by damaging them. The stress of lifting a challenging weight creates tiny tears in the structural proteins inside the muscle. These tears activate cells on the outer surface of the muscle called satellite cells. Satellite cells multiply and send cells to help repair the damaged areas. These cells donate their nuclei to the damaged fibers, helping them to repair by increasing the size and number of proteins. In addition, through a complex sequence of events, your immune system also aids repair by boosting the protein content of these damaged muscle fibers.
A diet adequate in protein is important too in growing and maintaining strong muscles. Consume too little protein (less than one gram per kilogram of body weight), and your body will counteract your strength training efforts by robbing protein from your muscles. However, consuming more than 1.5 grams per kilogram of body weight isn't beneficial either.
Finally, strength isn't just about beefy muscles. Learning proper technique and how to activate more muscle fibers at once increases strength. This is why newcomers to strength training will often gain strength rapidly at first. They're learning more efficient patterns of movement.