My brother (college student) watches my baby for $600/month. do i report this to IRS in order to claim dependent care credit? does he pay taxes? my mom claims brother

Yes, he must file a tax return and report the $7200 (assuming he worked for you all 12 months) as Self-Employment income on Schedule C and pay the related SE taxes.   When you enter your expenses, you need to enter his information as the care provider, including his SSN.   On his Schedule C, he is the owner of his own business, so the questions in the business profile section are asking about him and his line of work as a care provider.   If anyone else pays him to babysit or do similar type work, he would add that income to the same business (Schedule C) return.  

If he is a dependent, he is required to file a tax return if he has over $400 of SE income, so he does have a filing requirement.  If your mom is claiming him, make sure that he answers YES to the ''Can someone else claim you?'' question when he files his return.