1.Strength
Isn't that the main reason why you are doing this exercise? I assume it is because one-arm push ups are one of the top-level calisthenics exercises because they hit your chest and triceps very hard if you can do at least 10 in a row. You will also feel the BURN in your legs especially your groins.
2.Balance
You will also improve your body balance because it is hard to do one-hand push ups without having a full control over your body. To stabilize your body during one-hand push ups spread your legs and put your left or right (depending on which hand you are using) foot in the position in which your fingers are parallel with the floor and fully extended.
3.Fat loss
This exercise requires a lot of effort and muscle work which burns calories and leads to increased fat burn. Also the greatest feeling after one-hand push ups is the pump you feel when blood rushes to your muscles bringing more oxygen.