Imagine the minimum wage is $100/hour. There's a massive number of job which simply do not produce that much wealth per hour - they cannot exist, because to offer that job to someone is to lose money. All those jobs disappear.
Setting aside the stupidity of $100/hr minimum wage... (I mean, why not $1,000,000/hr right?)
The jobs that people do for under $15/hr still need to be done. Not every job produces wealth. Nobody gets rich by having clean floors, or mowed lawns, or bagged groceries. However, these are examples of tasks that arguable have to be done by someone, and the cost of not having them done can, at least in some cases, be argued to be greater than $15/hr.
The same applies to jobs that "do not produce that much wealth" - they still need to be done. Either you pay someone $15/hr to flip burgers, or you stop selling burgers and go out of business. Don't want to go out of business? Pay the $15/hr and increase your prices by the ten cents or whatever it averages out to be. What a goddamn stupid argument you're making.
I'd rather pay an extra buck for a trip to the local fast food place than have my tax dollars end up subsidizing the employees through food stamps and housing because they're barely paid enough to afford the same food they cook all day.