Short Answer: With the food God commanded Noah to bring onto the Ark
Bible reference: Genesis 6:21
“And take thou unto thee of all food that is eaten, and thou shalt gather it to thee; and it shall be for food for thee, and for them.”
God told Noah to bring all food that is eaten aboard the Ark for himself, his family and the animals. It should be noted that originally man and the animals ate plants [Genesis 1:29-30]. The fossil record reveals that many of those animals had become meat eaters before the Flood, but mankind was not permitted to eat meat until after Noah and his family left the Ark [Genesis 9: 2-3]. Creation scientists suspect that God caused meat eaters like lions and T-rexes to eat plants as they originally had when God created them, before Adam sinned and death entered the world. [See Why Didn't All of the Lions & Tigers & T-rexes [oh my!] Eat Everyone on the Ark? for further information]. So we suspect that “all food that is eaten” were plants.
While the Bible doesn’t tell us specifically how Noah and his sons fed the animals, Creation scientists have suggested that they used a system similar to a birdfeeder. Basically, the animal’s food dish would continually fill itself. Some animals may have hibernated and would not have needed food during these long periods of sleep.
Creationists have also entertained the possibility that meat eaters could have been sustained on dried fish, packed away in barrels.