Other posters have mentioned the increased solubility with increased temperature, So I'll mention another mechanism that is also at play when you're using soap.

This one might be a little more advanced, partly because I'm personally unable to put it in easier-to-understand terms

Molecules in soap have two 'sides'. At one end, are long chains where the molecules are nonpolar, like grease is, and at the other end is a highly polar region, like water is. Polar molecules attract to one other, so the water is able to dissolve and grip onto polar regions of the soap molecules. Nonpolar molecules also attract to each other, so the long nonpolar chains 'grip' onto the grease and stuff.

The bit where temperature matters is that nonpolar molecules attracting each other is in fact a process driven by entropy, the tendency of systems to go towards 'disorganized' rather than organized ones, all other things held equal. Entropy increases with temperature, so the strength with which soap molecules are able to hold onto grease molecules increases with temperature.

To elaborate upon the entropic nature of nonpolar-nonpolar attraction(hydrophobic interactions), the idea is that, in a system of nonpolar(like grease) and polar(like water) molecules, there are overall many, many more configurations of the same energy level that the system of molecules can take where the nonpolar molecules are in contact with each other and not with the polar molecules, than the other way around, so nonpolar molecules are more likely to be found together, rather than dispersed throughout with the polar molecules. Increasing temperature increases the energy in the system(by speeding up the molecules), giving access to more and more configurations, which decreases chances of nonpolar and polar molecules are in contact, which increases the strength of the apparent forces between nonpolar molecules.