The minimum wage is a government-imposed price floor on the commodity known as unskilled labor. (Of course there should be no government-imposed price floors on anything.) The goal of those advocating a $15/hr. minimum wage is to make the base wage for unskilled labor a wage at which one can live a comfortable, middle-class lifestyle. In addition to being hugely destructive to the economy as a whole, even the goal is damaging because it discourages advancement and self-improvement. Teenagers and new entrants into the workforce should 'pay their dues' while they develop marketable skills and earn (rather than demand) higher wages as their skill sets become better developed and more valuable.
Attempting to short-circuit the market mechanism for determining value (supply and demand) and instead imposing arbitrary price floors will only serve to create surpluses (unemployment) and drive up the cost of goods and services. (Incidentally, the increased price of goods and services will also impact those making the higher minimum wage.) Another important point is that your labor is not inherently 'worth' $15/hr. It may not even be worth $7.50/hr.! The real value of your labor is purely a function of supply and demand, and if there is not enough demand or if there is too much supply, your labor is worth less.
If the prevailing wage is insufficient for your needs you have essentially two options:
- Learn from the market signals and either move somewhere that your current skill set is in higher demand or develop a different skill set which is already in higher demand.
- Lobby for the government to force employers to pay you more than your labor is actually worth.
Those who value liberty will choose the first option. Those who advocate force and government coercion will choose the latter.