Where does MobaXterm keep its home directory?

You can see where the home directory is stored (and open it in Windows explorer) by using the following command:

open /home/mobaxterm

Go to Settings->Configuration->General and set the Persistent home directory to the folder of your choice. Then you will obviously know where it is.

However, if you have not done this before and you already have files in that directory and want to copy them out, try looking in C:\Users\your_username_here\AppData\Local\Temp\Mxt73\tmp\home_your_username_here. (assuming version 7.3, you might want to try Mxt7N for other versions like 7.N). After you've copied your files, then set a persistent directory so that next time it will be easier to find.

You can always use

open `pwd`

to open current working directory in Windows Explorer !

The simplest way is probably:

open ~