What Prestige Does?
Prestige determines what rank your country has, deciding if you are an (un-)recognized major/minor/regional power, etc, and thus how many interests you can have and in which you can then use diplomatic plays.
Prestige is the direct show of your projected state power.
Your combined Military, Navy, industry and social progress all add to your Prestige, and this in turn grants you the power to project influence zones to begin the process of claiming colonies.
In diplomatic plays, your prestige also translates into tokens for expanding on what claims you wish to make should war break out over the said diplomatic play. As Northern Germany, I now have roughly 100 tokens during diplomatic plays, meaning if someone does not back down I have the influence to claim the equivalent to half of Austria if I should so wish.
A Word Of Warning!
Claiming territory outright in such plays feels like a win, but will rapidly escalate how hostile other nations are to you. Some will even want your spot on the prestige ladder (Great Britain wishes to remain a super power of number 1).
You can find your reputation in your nations main screen by clicking on your flag (top left). It drops by 5 every year but it just means try not to go too mad for land. Puppet-ting states is easier, and provides money boosts, but nations might try and steal said puppets.