I don't need your approval. I just need you to tie the laces.
The joke is about an indignant horse insisting that they should be allowed to wear these unusual shoes, despite the cowboy's skepticism. The horse needs the cowboy to tie the shoelaces because its physically unable, but the cowboy is skeptical if the shoes should be allowed. It's funny in part because the horse is acting like a teenager asserting its independence, rebelling against a parent by wearing unusual clothing.