You don't mention what instrument you have, so here are my 'standing orders' for higher end Yamaha keyboards. Here is a YouTube video of a Piper playing. Presumably this is what is meant by 'Your fingers, however, should never take a breather.'

I have noticed that in this music there are 'grace notes' between every note of the melody, not just where two notes follow on that are the same. Here is some music showing how the piper plays those extra 'skipping' notes ('grace notes'). For this reason you have to do something to distinguish between two identical consecutive notes, and I think this is what you were referring to when you said you heard lots of 'trills'. On a bagpipe you can't make the same note sound twice, so have to 'skip' elsewhere and come back to the note to hear it sound again. Your fingers, however, should never take a breather. literally droning on by constantly sounding the same note.

You must get your bass to emulate the sound of the drone. So don’t allow your fingers to ever be idle. The piper emits a continuous, even sound throughout his rendition. The Chanter is the melody pipe, and the Drone just 'drones' on and on. The bagpipes consist of an air supply, a Bag, a Chanter and a Drone (sometimes more than one).