Minimum wage is supposed to provide for a basic standard of living. It is a voluntary system in which employer can find ways to employ fewer employees. Minimum wage also allows employers to highlight that a higher authority has dictated the value of low-skill labor, providing strong negotiating power to keep such wages low regardless of the work involved.
If we eliminate our welfare system--retirement (social security, government pensions), family and food security (food stamps, WIC, HUD housing assistance), unemployment, and a few other low-hanging fruit, but excepting medicare and medicaid (due to complex economic impact)--and replace it with a universal basic income set on a flat tax of 15% of all personal income (welfare currently represents 25% of personal income), everyone would be able to afford very basic housing and access to all basic needs. No one goes homeless, no one goes hungry.
Under such a system, your employment decision is between NOT working and living in a shitty apartment somewhat bigger than an RV (224sqft, with 6x9 bedroom and 10x9 sitting room, is my baseline), or working some hours for some pay. That whole employment thing is a costly impact on quality of life, but a doubling of your income--about $5/hr for a 40 hour week today on the numbers I've given--would bring your quality of life up sharply. The return diminishes as your income grows--$5000/year to $15000/year is big, but $150,000/year to $160,000/year is negligible--so the incentive to work is highest for the unemployed, but so is the power to not work if the terms (wage) are bad.
With no minimum wage, a very basic job is worth your time. Running a cash register in an air conditioned room is comfortable and non-taxing, and not worth much--enough to improve your quality of life considering the time you lose to employment. Manual labor in hazardous, hot, sweaty, dirty, distressing, uncomfortable conditions will not be tolerated without additional wages to offset the misery of your work. Minimum wage would establish that such a job--doable by any odd moron--is worth the same as running the cash machine at McDonalds, or perhaps very little more. Lack of minimum wage gives no reference and no guidance, so employers must convince each individual that the wages are somehow fair without appealing to a higher authority.