This is an English ‘old ditty’, or rather an old, short simple song. It is also good pronunciation practice for ‘f’ sounds. Please try listening a few times and then try to pronounce the words by shadowing just after the recording.
Felt is a type of material often used to make hats (see the picture).
Felt is also the past simple and past participle of feel which means to touch.
Therefore this rhyming ditty is a play on these words meant to amuse the listeners of whoever is singing or saying the rhyme.
Of all the felt hats I felt, I never felt a felt hat like this felt hat felt.