I really like ordering fajitas at a restaurant, partly because of how the ingredients are still sizzling on the skillet when the dish is served:
When food is served hot off the skillet, it tastes hot and fresh; it doesn't have time to cool down and become lukewarm.
In the same way, the musician talking in your quote was saying there was a recording studio downstairs from an office, and they could start recording something "the minute we’d finished it." That is, there was was no time for the idea to "cool off," and they could record it with the same zeal and excitement they had when their creative juices were flowing.
As someone who has dabbled in songwriting, I can understand why this would be nice to have. Sometimes I have a melody in my head that I will want to use, but, a few hours later, when I can finally start trying to capture what I was humming, I've already forgotten the tune! I've since learned to get out my phone and record enough of the melody that I'll be able to remember it later; that is, I'm recording the tune while it's still fresh in my memory. (Or, I could say, I'm recording it hot off the skillet, that is, very soon after it had entered into my mind.)
This usage of the idiom is a bit of an analogy. One might say the skillet is your brain, and the food is the idea your brain "cooked up."