Who are these two players, and why is this match significant?

For the past 2 or more years of competitive Melee, there have been 5 players known as the 'Five Gods' of the game. They were basically untouchable except by one another. There's Mango, the Fox/Falco player; Hungrybox, a Jigglypuff; PPMD, a Falco/Marth player; Mew2King, who plays Sheik, Marth and Fox; and "the beast from Sweden" Armada, the best Peach player in the world. Leffen is a very up-and-coming player, also from Sweden, who is perhaps Melee's best hope of ending the era of the five Gods

Why Melee, why not Smash 4?

Melee and Smash 4 are very different, and Melee still has a huge competitive following. In fact, competitive Melee has never been bigger, and keeps growing.

I've heard that Melee is boring because it's all Fox vs Fox, and this match seems to fit that stereotype

It's true that Fox is the best character in the game, but only when played perfectly, and he's the hardest character to play perfectly. The Fox-only thing is a complete myth. The tournament this video is from was won by a Jigglypuff, and the top 8 contained Sheik, Marth, Falco, Peach, Samus, Pikachu, and only one dedicated Fox player. Armada only switched to Fox in this match because Leffen had his Peach figured out.

Why's this match considered good (maybe spoilers)?

Both players suffer humiliating defeats but have the mental fortitude to keep playing their hardest

The commentators used lots of terms I didn't understand, how can I possibly get into this game?

There was a great thread recently for new players to ask these questions, and there's another one that is still ongoing for new spectators to Apex. Other than that there are great resources on smashboards.com, and the two Smash Bros wikis. But those threads will definitely get you started!

Where can I learn more about the Melee scene?

The documentary, of course

So what's Apex?

Apex is the largest community-run Smash Bros tournament (ie not part of a big corporate tournament like EVO or MLG). This year's Apex is set to be the largest tournament ever, with ~1040 Melee entrants, and over 800 Smash 4 entrants (with surprisingly little overlap between the two groups). This is the Smash 'World Cup', so to speak. There ould not be a better Smash tournament to watch if you're new to the game, and the top 8 for each game will even feature a special channel with commentary for newcomers. Here you can see the schedule for the weekend and where to tune in to watch your favourite game, and here is a trailer for the event.

I want in, how do I play?

Get a Gamecube + Melee, or get a Wii, which are more readily available, and it's very easy to modify your Wii to play a backup copy of Melee from a hard-drive or USB stick. From there use this list to find your local scene.

tl;dr:
WATCH APEX 2015