Who started the idea of giving out trophies as awards?

Answer 1

The Ancient Greeks and Romans who handed a laurel wreath to the victor in games.

Answer 2

On the evening of May 16, 1929, some 300 film industry figures and their spouses gathered in the Blossom Room of the Hollywood Roosevelt Hotel to dine on squab and lobster and hear Douglas Fairbanks Sr announce the first awards of the fledgling Academy of Motion Picture Arts and Sciences.


http://www.oscars.org/aboutacademyawards/history02.html
And of course the Trophy in general is a reward for a specific accomplishment, and usually afterwards serves as proof of merit

http://en.wikipedia.org/wiki/Trophy