In order to understand Huckel's rule and the concept of planarity, you need to start off with the basic definition of an aromatic compound. You really just need to make a simple internet search for either - aromatic compound - or - aromaticity - and a number of pages will pop up. So, anyway, what is an aromatic compound?
In short, it's a cyclic compound, similar to benzene that has delocalized electrons, that make it more stable and less reactive. The delocalized electrons are usually represented as possible resonant structures of alternating single and double bonds.
In order to have delocalized electrons within a cyclic structure, all the atoms in the molecule must participate with p - orbitals (the p - orbitals carry the electrons that are being delocalized, and form the delocalization ring, called a P system and can come from a carbon atom, or any other element, a heteroatom that is sp2 hybridised). Since all the atoms must have a p - orbital, they can't be sp3 hybridised.
If they were sp3 hybridized they could exist in several conformations, like the cycloalkanes for example and would not be planar, but because sp2 hybridization results in the formation of double bonds, and double bonds do not allow the functional groups to rotate around the C- atom, the molecule remains planar. Try to imagine this, visualize the double bonds not allowing the molecule to bend, or make models of benzene and cyclohexane and try to twist them, you'll see the mechanism that makes benzene stay planar.
Huckel's rule is based on quantum mechanical calculations, and I would advise you to just remember it's formulation for now.