If you want to study how a polynomial P(X,Y) behaves at infinity, you can't set eg X,Y=infinity, but you can take P(X/W,Y/W) when W goes to zero. Now clear denominators by multiplying by an appropriate power of W and you get a homogeneous polynomial in X,Y,W. You can now set W=0 directly. Scaling a solution cX,cY,cW doesn't change the original solution since (cX)/(cW) = X/W, etc.

I guess the important part is clearing denominators turns problems with division into linear problems.

"Scaling a solution cX,cY,cW..." you are in fact stating the definition of homogeneity here.