When you are training in a combat sport like boxing or in Muay Thai your body needs to conditioned. To throw kick and punches you need to have strength in your muscles. Calisthenics are an easy to create that strength, since most of what you are working with is body weight. Which means that you can preform them nearly anywhere.
Push Ups are the one of the best known ways to develop upper body strength. Your chest will feel the burn form doing push ups but also your triceps, shoulder and your backs muscles are engaged too. One of the benefits of push ups is that if you do enough, you can increase your muscles too. If you ever get tired of the standard push ups then, there are many variations; move your hands closer or wider, raise your feet on a platform higher than your hands, etc.
Squats are the a really good way to develop lower body strength ( And I mean body squats) . The three large muscles in the lower body are actively engaged: glutes, quads and hamstrings. With martial arts like Muay Thai the legs are worked directly with kicks. However you still need to have strong legs for boxing too. Each punch will be deriving power from pressing your weight against the ground as you punch. This means that your whole body weight will travel in a punch. Squats can help with you being to realize that.
Chin ups are good for both the back and the arms as well. Your back muscles control how fast you can bring back a punch. So doing chin ups will help with speed and keep your arms from not getting tried after throwing punches. Also it will keep you back in balance with your other muscles. There are variation for chin ups and pull ups are good once you think chin ups are easy. The only down side is that you will need to find a bar somewhere to practice them on.
So the three areas of the body that you want conditioned for marital arts are the Legs, the Chest and the Back. Working these body parts will allow you to develop power in your strikes and keep your body fit too.