Here are 3 reasons:
Full Body Workout. A standard push-up requires contraction of the muscles around the knee joints, hip joints, pelvis, and spine to maintain a straight line from your head to your feet. Combine that with activation of the muscles on the back of your arms, chest, shoulders, biceps, upper back, lower back, and legs, and you get full body workout in one simple exercise.
Versatility. Push-ups can be used to not just build muscular strength, but also to improve power (e.g. a “clap” pushup) and increase muscular endurance (e.g. doing X number of push-ups in 4 minutes).
Flexibility. By altering your hand and foot positions, you can change muscle recruitment
patterns and joint stresses of the push-ups – making the movement harder, easier, or simply stressing different muscles.