One of the biggest ways you can eat cheap while traveling is by staying at a hostel that includes a free breakfast with the price of the room. The meal can be anything from just toast and jam to an omelette made to order and everything in between.
As for your other meals, you can easily find everything you need to make a nice picnic lunch at the local market down the street. Prepare some sandwiches or, better yet, buy some cheap local delicacies and try them.
As for dinner, why not take full advantage of the hostel kitchen? Check for a shelf that has free non-perishable food items left behind by other guests and make a meal around it. Even better, why not make a meal together with other guests? Pool your resources together and buy in bulk - not only is it cheaper but preparing and eating a meal together is a great way to make new friends.