If I'm running a business, and my payroll increases 30% while sales remain flat, I have two options:
1) slow or stop hiring, cut staff, or even go out of business;
2) raise prices to reflect the increase in wage expenses;
I don't see how either of those outcomes is very good for poor people struggling to get by on $15 an hour, even though we can console ourselves that we've "done something" to help them. Now, I am, admittedly, not an economist - perhaps it will work, or at least, it will help. But I really have sincere doubts that this is going to do much to really change the dynamic at the low end of the economic spectrum. I think it will largely end up being a feel-good measure that well-off, well-meaning people can use to congratulate themselves about, while doing nothing to change the fundamental reality of low wages and poverty.
The example you give only applies to a subset of businesses. Most will not have to go out of business because they have to pay people at least $15 an hour. A few little mom and pop's maybe. But Walmart is not making that calculation, nor is Target or Costco. They can easily absorb the increase in labor costs, and their profit will go from ridiculous to merely astounding. Besides, people's wages are companies revenue. Studies have shown that increasing the minimum wage can stimulate the economy by enabling people to spend more. This is most effective at the bottom of the income range because those people will spend the increase rather than saving it. So increasing labor costs is not the only dynamic at play.
I agree that this will not solve poverty or income inequality. But it's a step in the right direction. Our economy is supposedly 70% consumer-driven. It makes sense then that increasing the buying power among those most likely to use it would have an overall positive effect.