Sharp, here, simply means to keep their mind working and enhance thinking, which means to make someone intelligent. Example: 1. Playing Chess is an effective way to sharpen your mind. 2. He practises solving mathematics mentally to keep his mind sharp.

When you sharpen a knife you make it more responsible. When you want to cut something, the knife is more agile. In the example that you are giving it's kind of the same. In the sentence, the subject wants to feel the pursuit to keep himself "sharp". If you want to change "sharp" with another synonym you can use "on top of their feet" " He wants to feel the pursuit. He thinks it's gonna keep him on top of his feet". Best of luck figuring out this tricky language

Sharp means alert, intelligent, responsive, mentally prepared. To stay sharp has lots of synonyms - to keep in shape, to be in top form, to be ready for action.