Could anyone tell me how I could thin out a piece myself?

Well, depending on how low you want the density to be, just natural wear and tear will thin out a piece.

However, if you want a quick fix, a thinning shears might help.

Someone suggested coating the knots with conditioner from the base (bottom side). This will loosen the knots and then gently pull the hairs out.

Could you explain "coating" a bit more?

Sure! Just apply conditioner to the base with your hands making sure you apply enough to soak the knots really well. Probably need to let sit for a few minutes to allow conditioner to soften the knots. Then gently pull out as many hairs as needed until you get the desired density. Rinse conditioner out. Base will probably need a good washing to remove any conditioner residue that could prevent base from adhering properly when you go to attach.
I have not tried this yet myself. I believe the idea was originally posted by JRob ?