You can fill a glass of water to just slightly above the rim without it spilling over the
glass. what property of water best explains this phenomenon?

Surface tension due to cohesion

Explanation:

Water molecules are polar meaning they have dipoles. The hydrogen end is partially positive while the oxygen end is partially negative. This is because oxygen with high atomic mass attracting most of the molecule's electron cloud to its end.

Due to this phenomenon water molecules act like little magnets that attract each other. The attraction of the molecules forms hydrogen bonds with each other. This is called cohesion (attraction of similar molecules).  Due to this, water is able to create a surface tension that acts like a ‘skin’ and can support some weight (including its own weight) without collapsing.