I've never understood the argument that minimum wage increases increase unemployment.
It is a simple math/statistics argument. We have an economy, where every day someone is thinking of starting a company. They look at their expected returns. These expected returns will (taken across the economy as a whole) form a Gaussian-like distribution. If the expected return is greater than zero, they will start the company.
You then increase one of the costs (can be taxes, can be labor, whatever). This shifts the distribution by some amount. The area under the distribution where expected return is greater than zero has now decreased. So fewer companies get started, which means fewer jobs created.
So in the short term, you would expect little effect - existing businesses are sunk costs, so they don't shed jobs quickly. But long term, fewer companies are started so fewer jobs are created.
People on minimum/low wages spend all of their money, and usually locally. (discussion about spending habits)
The key point that negates all of this is that the number of widgets in the economy has not changed, so you can't be making anyone better off. More concretely: let's say you have 100,000 people in the economy, and the economy produces 100,000 hamburgers that cost $10 each. On average, each person also earns $10 each. So everyone gets 1 hamburger.
Now Congress passed a law mandating a $15 minimum wage. And let's pretend that no one loses their job, because subsidies! OK, now you have 100,000 people earning $15. But you still only have 100,000 hamburgers, because that is simply how many hamburgers 100,000 people can make. So what is the cost of a hamburger? $15.
Most government mandated actions like this can only either end up in inflation ($15 hamburgers) or unemployment (1/3 of the people lose their jobs, and the remainder have 33% taxes to prevent that third from not getting a hamburger). Money is not value, it is a metric we use to measure value. Giving away money destroys the metric, and doesn't help the recipient.