Usually, on a live CD or Live USB, all modifications are discarded when you reboot.

The persistency allow you to keep your preferences and data even after reboot.

The data are stored in a special file called casper-rw (for Ubuntu) and overlay-USBLABEL-UUID (for Fedora). This is a completely transparent process for the user.

In this step, you have to choose the size of the data you will want to keep. Usually, 300MB is enough to install some few software but you can set more.