Here is the best way to see it, when you divide you are breaking it up into pieces, you have a cake and are slicing it into pieces.

So 1/4 is saying you have a cake and are slicing it into 4 pieces.

Now here is the big difference between infinity and undefined.

Consider e=mc^2

Mass can be converted into energy, but never destroyed (splitting an atom converts the mass in an enormous amount of energy)

So saying you want 0 parts is asking to cut the cake into 0 pieces, or essentially destroy the matter.

This is undefined because it is not possible.

Now as you go approach the limit of zero, you are cutting it into an infinite amount of pieces, the total mass still exists, its just in many, many different, equal, pieces.