It's all well and good focusing on all the good things AV has to offer, representative local politics, constituency link, more equality of people's voices...but this is also about why we should move from the current system.
FPTP has three main flaws that AV does not share (we all accept AV has flaws, though in general it's flaws are less severe versions than under FPTP.
1) FPTP discourages diversity/encourages unknown results
FPTP works great when only two people are standing. Add more voices in and it gets a whole lot harder to work out as fair. The more candidates that stand under FPTP with some kind of support, the more chaotic the election gets. As votes split further and further the actual number of votes needed to win the seat plummet. It's dangerous in one direction, those that gamble and get the support can easily end up getting elected on only a quarter of the vote if there are enough candidates with support standing.
But also, it means that if you are concerned as a candidate about splitting the vote you may not want to stand. Look at what happened in Oldham and East Saddleworth. Now granted the Tories ran a candidate to placate the local party, but they wished actually to not run at all to give the Lib Dems a better chance. They KNEW that under FPTP fielding a candidate would actually hamper the chances of beating their enemy.
When you have a system that says a winner can be crowned because of how low the threshold is taken, or because two parties are too similar and will gift a common enemy of the two a win, then it is time to move to a system that doesn't make such fatal democratic mistakes.
2) FPTP means unequal votes
Around 25% of the electorate in 2010 voted for a candidate that came 3rd or lower. They might as well have stayed home, it would have made no difference to the final 1st and 2nd place results. However, if they hadn't have "stayed home" and instead voted for the 2nd or 1st place candidate then suddenly their voice would have been more influential.
Under FPTP if you vote for a candidate that placed lower than 2nd then your voice was less instrumental in the result than those that voted 1st or 2nd. If, under AV, we whittle out all the unpopular candidates and end up with a final two, the only equal way to deal with people's opinion is to give them all an opportunity to decide which of those already popular final two are their favourite...equal voices and equal votes.
When your voting system is saying that if you are knowingly voting for someone that is not likely going to be in the top two that you better change your vote, or stay at home, something is wrong.
3) FPTP favours large unified minorities over more "picky" majorities.
Similar to point 1 above, FPTP has the effect of meaning constituencies with a strong but minority voice are more likely to win than those voices that generally agree with each other but find different vehicles of representation. Whether it is Labour and the Left Wing of Lib Dems, or Tories and the Right wing of Lib Dems, or Tories and UKIP, etc...if these local splits happen then the winner is not the popular choice, it's not representative of local views. FPTP allows strong minorities to steal elections, something completely obliterated by the introduction of the AV system.