The leg press is designed to work the quadriceps muscles. The quads are a large muscle, so you need to work them hard to stimulate muscle growth. As such you will also engage the gluteus maximus when you are pressing hard.  To a lesser extent the hamstrings, abductors and calves will also be engaged. The leg press is regarded as a compound exercise as it requires multi joint movement. Your hips, knees and ankle are all moving during the leg press. So, the muscles throughout the leg all get involved.