A simple explanation can mean very different things to different people. thesquaregroot's answer tackles this from the "simple but still technical" perspective. For me, simple means (borderline) non-technical.
A sponge construction is named after a sponge. Not per se the animal, but the derived device, that you use to clean a blackboard, or your kitchen counter. The sponge is capable of absorbing liquid and chalk, and afterwards squeezing it out.
Every time you use the sponge to absorb liquid and chalk (data) it will remember this action, it is forever in its history. When a sponge is squeezed, the outcome is a mix of all the sponge's history: the sponge mixes all its former input, usually resulting in a goo-y mash of chalk, with the original inputs unrecognisable.
Another analogue is a perpetual stew: a big stew that's regularly replenished with fresh ingredients. The outcome is never the same, since it depends on the ingredients added, on how many people eat from it, or the season and seasoning, and possibly other factors.
A sponge can be built into many things, among which a hash function: you use the sponge to absorb the input data, and then squeeze out just enough to form a hash.
Or you can use it very neatly for Fiat-Shamir transforms, e.g. https://github.com/dalek-cryptography/merlin/, based on STROBE: the messages to the interactive verifier are absorbed, and the challenges are simply squeezed out.
Or you can use a sponge as a stream cipher: absorb key material and squeeze out as much cipher stream as you need.
At that point, the physical analogy with a sponge is getting difficult: a sponge can only squeeze out as much as was absorbed, while a cryptographic sponge can continue to squeeze out as much as necessary, until the security possibly starts breaking down.
Disclaimer: I've tried to have this answer to focus on intuition, which could impede on technical correctness. If you feel like I should mention some incorrect analogy, please let me know!