Median income for most states is around $50k/yr.
Federal poverty level is about $25k/year, or $12.25/hr.
Minimum wage is moving to $7.35/hr.
If, instead, we moved minimum wage to $12.25, then one person might just barely survive, while a couple would be at median income.
By increasing the Federal Minimum Wage, we not only provide more individuals and families with the ability to pay their most important bills, we also provide them with more disposable income which they can use towards purchases of non-essential items.
Think of it as stimulus, or tax cuts, but at the level of the private sector. Instead of passing over the latte because you need to purchase gas in order to get to work, you can purchase the latte and a biscotti. Higher pay for everyone means they will purchase more.