What is a normal subgroup "morally"?

This question is inspired by a podcast I watched between 3blue1brown and Strogatz and in there, Grant talks about a Twitter thread where someone asked what a normal subgroup was "morally", and that thread has since been deleted.

I'm asking this because I think I have a fairly good grasp of normal subgroups in terms of them being the only subgroups that can be kernels of homomorphisms, and I think that feels intuitive to me, but I wonder if I am missing out on some understanding by not having a "moral" definition for it.

("Morally" in this context refers to explaining why something should/would be true, in the sacrifice of rigorous proof)

Podcast btw: https://youtu.be/SUMLKweFAYk?si=6FfmZurz4cnaTt3V