The labor market is missing one of the criteria of a free market, the ability of participants to decline a transaction if they only receive disadvantageous offers.
This is not a problem as long as there are more open jobs than applicants; market forces provide sensible price points (different for different skill sets) here. If there are more applicants than jobs however, prices fall quickly as there is no supply elasticity, and the gradient is basically given by the cost of the commute.
The two major ideas to counteract this market failure are a minimum wage, and a Universal Basic Income (UBI).
The minimum wage approach tries to change as little as possible. Competition for low-skilled jobs is still fierce, but at least the holders of these jobs are able to live in houses, get proper nutrition and possibly also have free time in which to improve their skill set, which will enable them to enter a different market segment, reducing pressure at the bottom. Upward mobility is greater when there is a bigger surplus in workers' wages after cost of living, because it allows for better learning conditions.
From a government perspective, we want everyone to be upwardly mobile, as we can expect that low-skilled building and manufacturing jobs to become even more scarce in the future, while at the same time, we need lots of specialists (to the extent that the US still has special visa for skilled workers).
The \$15 figure is pretty much arbitrary, but a full time (40h/wk) worker should come out at roughly \$28,000 before taxes, which is not great, but should be sufficient for rent, transportation, food and further education; at the same time the number isn't so high that it leads to competition with skilled jobs.
Going with \$8.26 would still avoid the "race to the bottom" in wages, but leaves workers nothing above subsistence level, diminishing their education opportunities and thus worsening the situation as more low-skilled workers enter the market than leave it, so unemployment remains long term and workers remain low-skilled until retirement, which is not sustainable especially in light of technological advance.
For completeness, the UBI approach simply corrects the missing part of the free market by allowing everyone to reject bad offers. This would be easier to police than a minimum wage as it removes the requirement to look at individual contracts, and just goes with equilibrium prices.