Please help me understand why "put up with" means "stand, endure"

Solution 1:

Let's look at the etymology.

An earlier (now obsolete) version of this idiom is just "put up", which dates at least as far back to this 1573 quote:

Al this I put up quietly.
_{Letter-Book of Gabriel Harvey A.D. 1573-1580}

The Oxford English Dictionary believes it is "[p]erhaps originally a fig[urative] use of" this sense of "put up":

To place in a receptacle for safe keeping or for future use; to store, stow away; to lay aside, put by; (in later use also) to pack or make up into a parcel, in a basket, etc.

This sense is still used in some regions of the United States ("south. and south Midland"), although I think most people won't be familiar with it. An example can be found here.

So, based on the etymology, you can think of the expression like you're bottling up your reaction and putting it away.