It's one of those things that seems to be taken more on faith than evidence: higher minimum wage means fewer jobs. But the evidence just doesn't seem to be there. It seems like it should so people assume it does, but cities, states, and countries with higher minimum wages than the US just don't generally show higher unemployment (there are other social welfare issues that change and probably do lead to fewer jobs).
This article discusses states and cities in the US, referencing side by side county comparisons. I wish I could find the link, but Austrailia has a much higher minimum wage than us, and pretty damned low unemployment.
Pure economics wouldn't get you here, but maybe a bit more thought does. Higher minimum wage means more impetus to work, plus more income for those who do. More workers making more money will then spend more money meaning an increase in the demand for goods and services, which must be provided by employing more people. Moreover, if you have a business you want to succeed, you can't just fire everyone because their wages must go up. You can't succeed with zero employees. Maybe you can pass on some of the cost to customers, but only in a colluding society can that be guaranteed to work. Most likely the people on top (owners, executives) will end up with less profits, which, I suspect, is the real reason that business leaders oppose raising the minimum wage.
[Note: I would actually guess* the lower profit thing to be an immediate effect in terms of total $ and long term in terms of % profit.]
*Not an economist, and not doing the maths...which in this case are hard even for economists--too many variables--which I further suspect to be one of the reasons why they all "know" that higher min wage reduces jobs: that's what their maths tell them.