How did "represent" semantically shift to signify "standing in the place of another"?
English borrowed represent from Latin repraesentāre, which meant ‘present again, bring back’, hence ‘show’. It was a compound verb formed from the prefix re- ‘back, again’ and praesentāre, source of English present. The notion of ‘standing in the place of another’ is a post-classical development.
Word Origins (2005 2e) by John Ayto, p 421 Right column. Ayto doesn't expound this semantic shift.