Sure. Solid question.
I think that we can agree that a business- any business- will pay their employees the least that they can. it's in their self-interest to pay the employees the lowest amount possible.
The Minimum Wage makes sure that business can't keep pay so low that employees can't live. That way the lowest skilled person can make enough money to eat and pay rent.
Now, usually, people who argue against the minimum wage stay stuff like "well, they just need to get higher skilled jobs!" And we all agree, yes.
But. If you're working 2 full time jobs during the weekdays and a part time job on the weekends, exactly when do you have the time to develop those skills? Or get an education.
Another thing people who argue against a minimum wage say is "well, they'll pay me less!"
Actually, they won't. They'll have to pay you more.
Lett's say that you're a burger flipper making $15 an hour. And I'm a widget maker that wants to hire people. But not regular any ol people. Make a widget means that you have to be really good at flipping things. And I don't want to have to train people to be good at flipping. I want them to already be good at flipping. So, burger flippers are the kind of employees that I want.
So, if I offer $10 am I going to get any burger flippers? Nope. But what about $15? probably not any with skills that I need. In fact, I'll need to offer at least $20 an hour. A full third increase. Because I need experienced burger flippers, not n00bs.
So, you, the burger flipper could stay making 15ish OR come work for me for $20. No brainer, you're going to come work for me. You put in your 2 weeks notice.
BUT now your soon-to-be-former employer is left with a choice- keep offering $15 and spend time training people AGAIN or pay their current employees more so they'll stay. Since it's cheaper to retain an employee by paying more- they're going to do that.
So, now the burger flippr makes more money and the widget makers make more money.
When more people are making more money, they spend more. A lot more. In Seattle and Washington DC, where there is a minimum $15, the economy has increased.