Licenses allows your weapon to do more damage, and some weapons give you crit bonuses.
There are several stages for each license, the number of stages varying depending on the mob license. The higher level the mob, the more stages it has, the more damage it adds, and the more crit. Each "stage" adds certain amount of damage to your weapon, for example, first stage adds 25%, second stage adds 30%, etc. You move through the stages (i.e. levelling up the license) by killing that particular mob.
For a full table, it's here:
http://www.fiestafan.com/wiki/Licenses
I may have got some minor details wrong since I haven't really used licenses much before but those are the general details.
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