Exert basically means to put forth effort to do something. For example, when you exert yourself in a workout, you can really feel the burn in your muscles.
The Latin verb exserere — with its derivative form exsertus, which is the source of our word — means to "put forth." That's where exert gets its meaning of physical force — consider that when you try hard at something, you put forth effort to do it, you push yourself. But exert can refer to other ways you "push": you exert your influence to convince others to think your way; when one nation exerts its power on another; and gravity exerts its force on us, holding us down on earth.