The Wikipedia article on minimum wage laws [0] has a pretty good overview of some of the most common arguments for and against. Really, though, the whole article is pretty interesting for a Wikipedia piece and illustrates the level of disagreement between economists over the effects these laws actually have.
Some jobs are viable at $8 /hr but not $15 /hr so the business have to eliminate positions to stay solvent. That means someone who could have taken the lower paying job now has no job at all.
Also, inflation. When everyone is making at least $15 /hr then that $15 means less than it used to. All costs of goods go up to match the new supply of "wealth". Burger flippers make $15 /hr but burgers now cost $10.
In more meta terms: manipulating the market is never easy if it's possible to do successfully at all.
My opinion is that having a minimum wage influences people to not make their own target, since an authority figure essentially deemed one hour of work is worth X dollars. Removing the minimum wage would make workers think for themselves how much they should earn and I believe that inclusion in their earning process would lead them to plan better, work better, and therefore earn more.
Essentially, I believe that when the government thinks for the people like declaring a minimum wage the people tend to avoid thinking about it themselves.
