Time traveling and a certain pair of goggles.

[Enkidoh]

Some major points here need clarifying:

I feel there is something far more to the Echo than just going on vision quests at inopportune times. As Lahabrea once said, "Very well. Let us show these mortals the true power of the Echo. The power to break down the barriers of existence!". So much is still not understood as to the nature of our echo. Krile's allows her to see traces of souls if using a strong enough foci, while Fordola's own artificial Echo allows her to see minor shifts in aether, effectively giving her tip off when her opponent is going to attack. The Sahagin Priest's echo in patch 2.2 allowed him to move his soul between bodies after he was killed by Merlwyb.

Of course the Echo is more than just visions, although being able to view another's memories as if you had lived them yourself is the most common aspect of it. Moose termed it as 'jailbreaking the soul' and that is a very good way of looking at it: it's a way of overcoming the normal mortal limitations where the mind is contained with matter - when you have an Echo vision your soul is touching another's, allowing you to access and even interfere within their soul (as in, their memories). Sufficent mastery of it allows the user to not just touch another's soul, but to ascend your own: this is what the Ascians have done, and what they also gave to the aforementioned Sahagin Priest, becoming a being of pure aether.

There are many other aspects of the Echo, such as being able to understand any language on Hydaelyn without having prior knowledge of it, immunity to Primal influence (as in, tempering), and being able to shield others from a Primal's tempering influence. Not every Echo user has access to all these abilities apparently: Minfilia could only view other's memories and couldn't understand other languages (as well as being especially sensitive to Hydaelyn's voice), where as Krile had visions, but was particularly skilled with understanding beasts. On the other extreme a minor npc in the Waking Sands in 1.0 mentioned he couldn't view memories but was particularly potent in understanding languages. And the player is probably the most potent in the Echo of anyone.

It also goes hand-in-hand with the Blessing of Light and manifesting Crystals of Light, but that's another barrel of moogles...

Ours seems to (from a gameplay standpoint at least) allow us reattempt fights after we have been defeated by allowing us to slip back into an earlier moment in time before the battle. Giving us (the player) an idea of what to expect as we have already tried and failed to do.

Okay, this needs to be addressed - that is not the Echo at work but merely a gameplay mechanic that has no story connection. As far as the narrative is concerned, you only ever go through a dungeon/trial/raid/battle once and once only. A willing suspension of disbelief is needed here. What's important is that you only ever do a battle once for story purposes (except when it's a HM dungeon where it's specifically a second time you're heading through, hence the changed situation as you had already gone through there once before). Things like Squadron command missions handwave that away however. And ignore the 'Power of the Echo' buff given for failing duties - it's merely flavour dressing with little to no actual lore connection at all.

It's also shown in 1.0 (even if it was retconned, and no I don't buy that in the slightest as that is the easy way out that creates even more problems than if it was left in) that it is possible for another echo user to discern if someone is present in a past event they were not part of by using their own echo. It's how we joined the Path of the Twelve in 1.0, and that is how Minifila recognized us in ARR when no one else did.

That's because another Echo user is merely accessing the first Echo user's own memories and so can see for themselves they have the Echo too. But certainly those particularly sensitive to aether can also detect when the Echo has been used on them - this is why Raya-O-Senna realized you had the Echo, she noticed you had inserted yourself into her memories!

As for joining the Path of the Twelve, that was merely because Minfilia had specifically sent out special 'Watchers of the Path' deliberately to look for anyone showing signs of being awoken to the Echo (such as having visions or at the very least kept muttering doggerel about events that had happened ten years previously like the player did), so they already knew what to look for. The Watchers didn't seem to have the Echo themselves, but were just Echo 'talent scouts' if you will who knew the signs of someone awoken to the Echo's power and would then instruct them to head to the Waking Sands. In ARR the Archons took over this role when they noticed the player showing the same signs.

I suppose at this point i'm just fueling my own head cannon in an attempt to explain away inconsistencies (dreadful I know). But it also includes a way for ideas and information to be passed back in time. Much like Squall does in FFVIII. (again a big influence on the current raid)

The Echo is indeed very similar to Ellone's mental-time-travelling power in FFVIII, but in many more ways it is completely different, not the less because it's not actual time travel at all but merely viewing another's memories as if you had lived it yourself. And besides, Minfilia herself in both 1.0 and ARR summed it up best that no matter what you do in the Echo, it is not affecting the actual past because it cannot be changed in the Echo for that reason:

Minfilia: "The past is like a tablet of stone. You cannot uncarve what is already written."