For the first time since really Cataclysm, there's actually a world-spanning launch event for the Legion expansion. Given that this is a massive invasion of the Burning Legion, it wouldn't make sense for all the action to be stuck in the Broken Isles, so once 7.0 hits, we're going to start seeing some demonic insanity raining from the sky over several old, familiar zones.
The world map will show which zones are being invaded, and you'll be able to head there to help deal with the problem. So far, I've defended Westfall and Hillsbrad Foothills, both on Alliance characters. It would seem that while the invasions are going on, the faction guards of the various settlements are happy to have whatever help they can get, even if you are from the opposing faction.
The invasion goes in multiple phases. First, the major town (in these cases Sentinel Hill and Tarren Mill) will be bombarded by a Legion vessel - basically a Star Destroyer. You can actually fly up to these things, but getting too close will knock you back and deal a bunch of damage to you.
Once you and other players have defeated enough of the invaders coming with the bombardment, the zone's full invasion will begin.
Throughout the zone, boss-like enemies will spawn, along with tons of soloable demons and several demonic structures will embed themselves in the ground. It's actually pretty amazing to see - they pretty seamlessly get integrated into the environment. You'll need to down these bosses and take out the fel crystals that support these buildings. Once enough of the invasion forces have been turned away in this manner, the invasion's boss will spawn in the main town, and you'll be able to defeat them to end the invasion.
You'll get Nether Crystals as a currency reward for these events, along with occasional loot boxes at the various stages of the invasion that seem to drop 700-level loot. There's a group of Illidari stationed near the town (at least Tarren Mill - I didn't see the one for Sentinel Hill, but I may have just missed that,) who can sell you a number of things including a battle pet, some transmog armor sets, some gear, and an item that will upgrade the gear you get in the invasions.