"Crutch on her" sounds like "crush on her". To have a crush on someone is to have romantic feelings towards another. When someone's leg is broken, they are typically given a cast for their leg, and crutches to help them move about. A crutch is a long stick that helps support the body. So, the joke could be taken literally, as in this person's date is literally bearing the weight of a crutch. Or, because "crutch" sounds like "crush" the joke is saying "my date could tell that I had romantic feelings for her". The joke is playing on the phrase "crutch on her" because it sounds close to "crush on her".
I went to prom with a broken leg . During the slow dances my date could tell that I had a crutch on her .