People who support the policies of parties other than LibLabCon often don't actually vote for the 'minor' party because they don't think the party can win, so their vote will be wasted.
Under AV everyone's 1st preference will be a true, honest reflection of whose policies they really support.
Under AV Labour and Conservative will get far fewer 1st preferences than they do FPTP votes. LibDems will get about the same, and the (so called) minor parties will see their true level of support push their 1st preference counts way above their FPTP vote count!
On its own this will not give 'minor' parties any MPs - but when the public see how little support those running the country really have, then the UK will have a revolution in democracy - and everything will change for the election afterwards.
AV will probably only survive one election, then full democracy kicks in - it maybe AV+ or it may be something else, but never again will 100% of the UK population be governed by a party with only 25% support.