There's an article on ABCnews which gives a bit more detail. Specifically (emphasis mine):
Reps. Tim Ryan and Jan Schakowsky joined former Ohio Gov. Ted Strickland in taking the challenge–and are each living on a budget of $77 for the week – the same amount that a minimum wage worker typically has to spend on food, transportation and day-to-day expenses–after factoring out major costs such as rent and utilities.
Since minimum wage for a full-time job is $290 per week, they're working under the assumption that there is $852/month to pay taxes, rent and utilities. This is part of the Live The Wage Challenge, which claims an average housing budget of $176.48 and an average tax of $35.06.
I dig further into these number below the line.
According to eHow (a questionable source, but comprehensive), a single person will pay 10-15% of their minimum wage pay for taxes, depending on whether they get at least 29 hours of work each week.(1) Social Security and Medicare are are another 6.2% and 1.5% respectively, or $35.96 + $8.70 for the full time job. Assuming the 15% tax rate, that's a total of $131.66 out of the bi-weekly $580, or $263.32 out of $1160.00 per month.
So, to recap: $1160.00 monthly gross pay - $263.32 federal taxes = $896.68 as a base amount each month to pay state taxes, rent, utilities, food, and so on. The proposed $77/week works out to $308 of this, leaving $588.68 for state taxes, rent, and utilities.(2)
The official numbers for the challenge are $705.92/month for housing and an $140.24/month in taxes. Working backwards, that's 12.1% in taxes (I calculated 22.7%). I can only assume that there are tax credits that I'm not looking at - their number is supposedly a real-world average (although they have no references to support it), and mine is just theoretical.
1) This is based on a "$417 in 2 weeks" breaking point for tax brackets, and involves a bit of rounding. At minimum wage, you cross into the next tax bracket at just over 57.5 hours in a two-week span, or approximately 28.76 in each of the two.
2) This currently unanswered question on Skeptics.SE asks whether the $77/week is even accurate, but for purposes of this answer, I assume that it is.