The cells resume their cell functions almost immediately after thawing. You'll hear the term "media" a lot when talking about cell culture. Media is a general term for liquid food for cells. It contains macronutrients (carbs like sugar, protein usually in the form of amino acids, and fats), micronutrients (vitamins/minerals), some other components (hormones/growth factors), and a buffer to maintain a physiological pH.

There are a ton of different types of media for different cells but the most commonly used is DMEM (Dulbecco's modified Eagle's medium) supplemented with 2-10% fetal bovine serum (FBS) for additional nutrients. Often researchers will also add antibiotics such as penicillin and streptomycin to prevent bacterial growth.

The media provides the fuel while atmospheric dissolved oxygen provides oxygen. Once the cells return to 37C, they spend a little time recovering as generally cells undergoing stress such as temperature changes and exposure to organic solvents like DMSO will stop dividing. This recovery period can take between 2-48 hours depending on cell type. For commonly grown cells like HeLa (from Henrietta Lacks) or HEK293, they take about 6 hours to recover and begin dividing.