On the subject of the minimum wage, people seem almost immune to objective, empirical arguments, but one has to state them anyway, if almost only for the record:
Wages are the price of labor. Higher skilled labor is worth a higher price. Lower skilled labor is worth a lower price.
When you set a legal price control on labor, which is what a minimum wage is, you make it illegal for low skill workers to sell their labor. It's as simple as that.
People generally seem to understand that prices reflect value, except when it comes to labor.
What if we made a law that said fast food hamburgers could cost no less than $20? People would be outraged, and would (correctly) argue that this made it impossible for low-income folks to afford fast food hamburgers.
Well, if you pass a law that says people who work in fast food places can make no less than $20, you make it impossible for low-skill workers to sell their labor to the fast food place. The owner of the business must respond to the price control, just like the consumer must respond to price controls.