It causes the item to take more affixes than its legendary potential, e.g. taking three affixes with a 1LP unique.

The chance depends on the chance for that unique to drop with more legendary potential than the one you're using. For example if you're using a 1LP Calamity the chance depends on the chance for Calamity to drop with 2 or more LP at level 100.

It's there to inject some more RNG back into the system to make it more exciting, but also to make the other affixes on the exalted item still matter when you're using a 1LP unique and can guarantee the one you want the most.

So crit success doesn't just add a single LP, it can potentially add 2+ LP? You can crit success on a 1LP and turn it into 4LP?

You can, it's just really unlikely.

Character level has no effect on it, but area level does for drops. It's (area level + 20 - effective level for LP) that's passed into the LP calculation.