The double bench explosive push-up is one of my favourite exercises for developing upper body power. You do not need to use a barbell or dumbell to develop a tremendous amount of power in your upper body.
Unlike some barbell or dumbbell exercises where your body is in a fixed position, for instance during the bench press, in order to execute this exercise, your entire body must be working in unison, including your core and glutes. This will provide your pelvis and spine with the stability they need so you can generate a lot of power with your upper body.
Obviously this exercise is very advanced as I am using high benches, so some regressions would include: hands elevated explosive push-ups on two benches (do not drop down to the floor), clapping push-ups on the floor, and then these double box explosive push-ups but with lower boxes.
Here are some key points to this exercise:
1) Set yourself up on the benches so your shoulders are directly above your hands and your body is in a straight line from your head to heels.
2) Right before you descend into the push-up, take a deep breath into your belly (360 degrees of air around the spine), brace your core, and squeeze your glutes. This will provide your body with the stability it needs so you can generate a lot of power with your upper body.
3) You can do a bit of a hip hinge as you are exploding up, but do not allow your back to hyperextend or rib cage to flare. Actively tuck your rib cage towards your pelvis and keep it that way for the duration of the set.
4) When you hit the ground, make sure that you keep your shoulders packed and directly above your hands.
5) Reset before each rep.