In the intuition behind classical set theory, a heirarchy of sets is built up from the empty set by an iterative process in each stage of which new sets are formed from sets formed at earlier stages in the process.
This intuition makes us comfortable to assert the non-existence of problematic sets whose existence may lead to paradox.

Since this intuition is an important element in our confidence that set theory is consistent, it is natural to want to see this stated in the axioms (even though we may not need to know this to make use of the theory, and even though adding an axiom can only increase the risk of inconsistency).

In first-order axiomatisations of ZFC well foundedness is expressed by asserting that every set s has a member which is disjoint from s.
This formulation has two disadvantages.
The first is that the connection between this axiom and well-foundedness may not be obvious to all.
The second is that it doesn't guarantee well-foundedness.