Okay so I couldn't find much about triggering an invasion, besides killing about 500 mobs. I did get one to trigger, however, and I figured I'd make a post here with a couple of useful tips that I would've liked to know.
1) My farming location was Lost Veil Anzu, which triggers a Botani Invasion. My reputation with Arakkoa Outcasts went from ~11500 Honored to ~4900 Revered. I am a Human, and about 2000 of that reputation was with a Standard of Unity down (+10% Rep). I believe that all mobs there give 5 base reputation a kill (before rep bonuses), so if my math is right, that'd be about 900 kills? Correct me if I'm wrong. I did not do a Botani Apexis daily today, so that shouldn't be a factor. I believe I've completed two or three total before today's weekly reset. Unknown if the Tuesday reset plays a part in resetting your invasion progress.
2) When the invasion triggers, the Garrison Report minimap button will glow (like a mission has been completed). At the top of the Garrison Report in the middle will be a big yellow " ! ". Hovering over it displays a tooltip stating that your garrison is under attack. This doesn't mean that you need to rush there to defend it, however. It begins at your Sergeant next to your Garrison Cache (for Alliance, at least). He'll have a quest for you, then you must speak with again to start the invasion.
3) The invasion itself is quite easy. Gold requires 1000 points. Different types of enemies will spawn, killing them will reward you a number of points. More difficult enemies award more points. It starts off slow, but quickly ramps up to lots of enemies spawning. It does not require a particular group to complete. Though three 630+ DPS would be ideal if you're aiming for gold, as spawn rate is related to how quickly you kill enemies. After ten minutes of killing, enemies stop spawning, and you clean up the remaining ones. You'll be awarded points based on how many buildings are alive, a morale bonus (not sure what that is), and a boss will spawn, who also rewards points. My group had 1000 points after the ten minutes were up, but ended up at about 1250 when the boss was dead. So don't worry if you're a little shy of 1000 points after the ten minutes, unless your garrison is up in flames.
The rewards are Invader's Abandoned Sack for Bronze. For Silver, Invader's Damaged Cache. And Gold, Invader's Forgotten Treasure. Achieving Gold rewards you with all three bags.
If you're going for gold, it's not worth it try it solo. I don't imagine solo gold will be possible until Highmaul gear becomes available. It's easy enough to find people for these right now as not many people go out and trigger them, and LOTS of people just wanting to jump in for the free loot without having to do the grind.
I hope this post answered some of the questions regarding garrison invasions. If you have any other questions, feel free to ask. I may be able to answer them.