Paying a public figure provides funding for their campaigns. By paying to keep them in service, they imply that they want to receive some sort of special favor in return. Unfortunately, even those who are elected to serve are just as easily susceptible to bribes and coercion as everyone else in the world.
Most of the time, public figures who are getting appearance fees are speaking to sympathetic audiences and at sympathetic locales. It's reasonable to compensate some of them for their time and travel to attend a commencement or a dedication. People want to see public figures since they know them and unless something shady is going on, an appearance fee isn't a bribe.
No, being paid to speak does not constitute bribery. Just because a person has been compensated for a speech does not inherently mean the content of that speech must be altered. A person who is being paid to speak may choose to say things that upset their host if they find it necessary to their message. Money is not intrinsically corrupt, it is the compromises a person chooses to make for that money that leads to corruption.
It is not bribery. This is because public figures have accomplished some kind of feat -- whether it is achieving fame by acting, winning in sports, or corporate leadership -- and people want to listen to their stories and inspiration. It is more like a "learning by telling" method, only for pay.