What Determines The Amount Of "Current" In An Amp?

The majority of Class D amps use a Switched Mode Power Supply (SMPS) design. SMPS are also sometimes used with amps in other classes of operation like Class A/B. SMPS designs do not require large transformers as conventional linear PS designs do, since they run much more efficiently. Class D output stages (independent of whether they use an SMPS or linear PS) are also very efficient, so they do not require large heat sinks that other designs do. Class A biased output stages are the most inefficient and therefore require an extremely large amount of heat sink area.

Everything is relative, and it depends upon the specific design. One example which uses the Hypex nCore 1200 modules is the Merrill Audio Veritas which is rated at 38 amps max current output, and 400w into 8, 700w into 4, and 1,200w into 2 ohm loads. It isn't clear how long it can provide that amount of current before thermal overload/shutdown.