Mostly if something is not constant in volume/level and it keeps going up and down throughout the entire mix, then you can use compression to fix that problem and keep the volume constant. Sometimes you might use a compressor to bring up the attack of a sound and bring down the sustain, while sometimes you might want to bring up the sustain and bring down the attack.
Compression can also be used to glue sounds and make them sound as one, for instance, using a compressor in a drum/vocal bus or group channel will glue the sounds together. Sometimes you can use a multi-band compressor to fix a specific problem frequency.
In some cases you might find that using an EQ (static EQ) might make a sound dull in certain parts of a mix (maybe in the chorus because there's too many sounds) so using a multi-band compressor to squash the frequency (in a good way) whenever it exceeds a certain threshold will fix the problem a lot better than using an EQ. Those are the most basic or common uses for compression, hope it helps.
In the case of your song/remix, I think your friends mean you should use compression to control the loud peaks of the leads. I can hear that you sidechained the leads so some minor compression should fix the problem.
