Looks like you're learning about diffusion.
Diffusion is the net movement of molecules down their concentration gradient (high to low concentration).
The dye inside the dye bottle contains 100% concentration of dye.
If you put the dye in hot water, the dye would spread, or diffuse, pretty fast throughout the water, from its high concentration to low concentration. This is because hot water increases the energy of the molecules, causing them to move fast in random directions. The hot water would make both the dye molecules and water molecules move fast so the dye can diffuse in the water until it reaches equilibrium faster. You would see that all of the water is colored faster than the dye in cooler water.
Normal cool water would make the solution reach equilibrium slower than the hot water.
