chroot isn't strictly necessary for everything. If you just want to edit a file you can usually do so without it.
Some system tools expect the system to be laid out in a particular way. Things like package managers (rpm, yum, dpkg, apt, apk...). There's no way to tell these commands that your system is laid out in a different way.
So let's take a real example like Debian's
dpkg. It keeps track of installed packages in
/var/lib/dpkg. When you boot into a Live USB and mount your broken system to
/mnt dpkg's library will now be in the wrong place... it will be in
/mnt/var/lib/dpkg. You can't tell dpkg to use this, so you can't use dpkg to fix your broken system unless...
chroot /mnt you shift where everything is. So
/mnt/var/lib/dpkg goes back to
/var/lib/dpkg where it should be and so dpkg will function again correctly.
No you don't usually need to add the swap. The layman's description of swap is it uses some HD space to extend your RAM. For this type of activity you usually only need a fraction of the RAM you have. If you were about to run out of RAM you can add it at any time... So if SWAP is mounted from a file
/var/swap you can either
After you chroot.