A bit of a recap here...but it's not good enough to pretend that AV elects the least worst candidate.
If three candidates, Tom, Wendy and Kim stand for election, and get the following shares under FPTP, then Kim would win...
If Tom never stood, then under FPTP Wendy would win, assuming that half of Tom's supporters vote for Wendy, and half don't bother to vote at all...
If the first election was run, we'd likely say Kim was the most popular (or held the most votes), whereas if the second election was held we'd have said Wendy was the most popular.
Using FPTP's very own definitions then, it's clear that AV round 1 would result in the first set of shares, and then result in the second set of shares. Wendy would rightfully win as the more popular candidate...but more than that a candidate with majority support and true endorsement from the local area.
This process is especially important when you consider perhaps that Wendy is a Labour candidate, Tom is a Green candidate and Kim is a Tory candidate. The population (as the second result shows) are clearly leaning towards green-left ideals. Under FPTP, with all three candidates standing, the population (who are clearly green/socialist) would end up with a conservative candidate.
So AV re-runs FPTP elections to eliminate vote splitting that can allow a candidate that doesn't represent a constituency to win, and it does so by electing the most popular candidate just as a FPTP election would. A fairer system for better democracy!