Quality Score is Google's 1–10 relevance rating of your keyword-ad-landing-page combination — and yes, it matters, because it directly discounts or inflates what you pay per click. High relevance can win a better position at a lower price than a sloppier competitor's bid.
The practical levers, in order: make the ad's headline echo the search, make the landing page's headline echo the ad, and keep ad groups tight enough that one ad can genuinely match its keywords. That message-match chain is most of the score.
Don't chase 10/10 as a trophy — it's a diagnostic. A 3 tells you the searcher, the ad, and the page don't agree about what's being sold; fix that and the score follows.