Answer: My first impulse is to say, you just feel around until it sounds right, but what does that mean? The thing is to find the tonic. That's the "home" note. Assuming that this person is singing tonal music, i.e. any popular song, then there is going to be one pitch that seems like home and to which the singer often returns. In your imagination it will be the ending pitch, the one that wraps it all up. Songs don't always end on the tonic, but they seem like they ought to. So you get into your head what sounds like the home or tonic for the song you're hearing.

Then you want to hope that the singer is in tune with the piano and didn't happen to choose a starting pitch that was in between two keys. If the singer heard something from the piano or another tuned instrument before getting started that should be OK. If you are one of those with absolute pitch then you'll know right away the name of the tonic and can put your hands in place. But if you instead have relative pitch, like most people, then you'd probably need to quietly run over a few notes to find the singer's tonic.

Once you have the tonic it's just a question of major or minor. That's beyond the scope of this answer, but you probably know what minor sounds like. Most popular songs are in major keys, but if the scale is minor then you'll know what to do - I expect you wouldn't be asking this question if you didn't already know how to improvise in a key.