Raising the minimum wage form the current $7.25 an hour to $15 an hour is calculated to raise the cost of fast food for example by 4.3%
If some segment of the population suddenly has twice as much income, and maybe 5x as much disposable income, that's going to put upward price pressure on lots of goods and services.
For instance, the people who are living with 3 roommates each making minimum wage now decide to get a bigger place, or just have 1 other roommate (maybe a GF/BF). Suddenly demand for housing goes up. The people who used to compete for low rent places in crappy neighborhoods are now competing for medium rent places in decent neighborhoods. Now the manager, who lives in a decent neighborhood, faces a rent increase and wants a higher salary. Did your 4.3% include that?
I think people who think the minimum wage doesn't have a big impact are missing this key idea. It's all relative, and it's not just about direct costs. It's about, if I make 4x minimum wage right now, and suddenly I'm only making 2x minimum wage, that hurts me in many small ways that add up. Maybe these poor people start having more kids, and my kids' school gets crowded, and there's a bond referendum to build a bunch of new schools and hire teachers, and my property taxes go up. Maybe poor people stop taking the bus or walking to work and buy cars, and now there's more traffic, and the city/county/state need to add lanes to a bunch of roads, and there's a tax increase to pay for it. Now I'm being affected even if I don't eat fast food.
It's a selfish viewpoint, but really it's no more selfish than those who want a higher minimum wage.