Struggling to understand why the analytic-synthetic distinction is very important Well let's have a little go at this. Language works by means of distinctions: beer is distinguished from not-beer, otherwise we don't know what we are talking about when we say "beer". and that would be a tragedy, because it is a useful and important distinction, even if it becomes blurry at the barleywine edges.
One of the distinctions that philosophers find useful and important is between the word and the thing, sometimes called the signifier and the signified. The convention is that when one wants to talk about the word "beer", one puts it in quotes, and when one wants to talk about the drink beer, one does not. Thus there is a clear difference, beer is a nourishing drink, whereas "beer" is a word.
So now, philosophers of beer can discuss the defining (necessary and sufficient) features of beer, What makes something beer and not a rabbit? Cue much talk of hops, barley malt, fermentation, and the amount of froth on top. Does it have to be liquid, or is a frozen beer still a beer? Are the yeasty dregs at the bottom of the barrel beer? All this talk is talk about what the word "beer" means, about what counts as beer, we are trying to sharpen up those blurry edges of the distinction between beer and not-beer.
So it is the case that(P1.) Fosters is not beer, but the recycled piss of inebriated Australians, and this is a matter of fact, given our shared understanding of what "beer" means. And this is what we call synthetic proposition, because it turns out that Australians also make proper beer that they do not export, but wisely drink themselves.
However, it is based on not only the facts of the case, but also the analytic proposition that (P2.) the recycled piss of inebriated Australians is not beer. This is analytic, because it is not about beer, but about "beer". The facts of the case - that Fosters call their drink "beer", are not decisive, because to most philosophers of beer that is simply an abuse of language.