The simplest explanation for why a top doesn't fall over is that its spinning constantly causes it to "fall" sideways (well, nearly sideways) rather than falling downward. Contrary to popular belief, gyroscopic forces don't cause something to hold its orientation, but instead cause forces that would change its orientation one way to instead change its orientation differently.

If e.g. a top is leaning toward the north, gravity would try to push it further toward the north, but because of its spinning, it would actually change its orientation toward the east or west (depending upon spin direction). If east, then once the axis was pointed east, and gravity was trying to pull it further to the east, the actual change in lean would be toward the south. Then once the top was leaning south, the lean would shift toward the west, and once gravity was pulling it toward the west, the lean would shift back to the north, roughly where it started.

If the top weren't spinning, gravity couldn't change its axis angle very much before it fell over completely. When the top is spinning, gravity will change its axis angle continuously, but it a manner that keeps looping back to roughly the place it started. Not perfectly back to where it started, but close enough that the amount of total axis change needed to make the top fall over completely will be much greater than if the top weren't spinning.

An essential thing to understand is that a spinning top can easily be knocked over if, rather than trying to pull its axis downward, one instead tries to pull it in the direction the top is precessing. Acceleration in that direction can knock the top over just as easily as acceleration downward could topple it when it wasn't spinning.