While like-dissolves-like is a good principle to follow, what's happening here is a bit more complicated than that. Here, you basically have three components. Your sample, the TLC plate, and elutant. So basically, you load your sample onto the TLC plate, which basically means that the molecules in your sample associate with silica. How strongly they associate will depend on their properties, such as polarity and H-bonding. Now, what happens - and this is the principle of TLC and silica column separations - is that your elutant comes in and competes for binding to the silica gel. If your sample is only loosely held, it will come off and travel up the plate. If it's held tightly, it will stick on until the elutant becomes too strong and washes it out. So basically, no matter whether your sample is polar or not, a strongly polar solvent will always move it up more than a non-polar solvent like hexanes. This is because with a strong, polar solvent, all your sample come off immediately and travel up the TLC plate.
The difference with using different strength elutants is to maximize your resolution. If you start off with a strong, polar elutant, everything in the sample will come out and will travel more or less together. You won't have separation of spots.