Knowing/WMG

The Watchers are, if not evil, at the least apparently amoral.
Consider the following: Early in the film there's a discussion on determinism versus 'randomness.' The thing is, in modern physics, quantum physics to be precise, there are certain things which cannot be known deterministically, such as whether an electron will show up with up or down spin in a certain case, or whether a photon will pass through or reflect off a glass pane. These random events cannot be predicted, and thus create uncertainty in the future. However, things can be predicted to be possible, and even probable.

What if, rather than the saviors these apparently angelic beings try to appear to be, they are instead looking ahead, seeing a possible way that things could come about, and then ensuring they happen that way? They knew there was a possibility that the sun would fry the Earth with a solar flare. Let's not get into how ridiculous that is, for starters. But since that flare was caused through a series of quantum events, there was always a chance it would happen earlier, or later, or aim in a different direction. Any number of things could have happened differently.

Yet, these Watchers apparently let the holocaust of Earth happen, rather than prevent it, and they only took a small sample of genetic sources off the planet. With a knowledge of quantum mechanics that advanced, they could have stopped the solar flare from hitting Earth in the first place. Instead, they apparently decided that the way they predicted was the only way things could happen, and forced it to happen that way. Nice job breaking the Earth, 'Angels'. Or maybe they're not what they seem to be.


 * Trust me, you aren't alone.

The prophecy and the arrival of The Watchers are completely unrelated.
I didn't check The Other Wiki, so I don't know whether this was already denied by Word of God. Now, let's come to the point. The girl prophesied the shit will hit the fan, and where to go to get saved (possibly knowing it'll only help her granddaughter and new buddy of her). Now jump forward these forty or how many years. Some aliens discover Sun is just about to go KABOOM: "Holy sh*t, what do we do!?" "Well, we ought to help these creatures... Find them a nice new planet or whatnot" "How? We don't have enough ships!" (for whichever reason) "So let's just take, like, some of their younglings, and yeah...".
 * It seems that the aliens/whatever were the ones who communicated the prophecies to her - she was hearing the same sort of "whispering" the two kids did. And really, how small is the spaceship that they can't take the kids father along?
 * Remember how Nichola Cage's character drank in the first part of the movie. Clearly the aliens took an invisible scan of the father's liver and decided that he wasn't long for this world anyway and rejected him.

There is no free will
The story takes place in a universe like Dr. Manhattan's. Knowing things in advance cannot change things. Nicholas Cage's character knows about 2 disasters in advance but is unable to change the number of deaths in spite of the fact that his disruptive presence on the train should have had some random effects on who lived and who died. The angels/aliens did nothing because they could do nothing: they can see their strings. We are all just playing out roles established at the beginning of time, and all we can do is accept. Maktub: it is written.

The Watchers are actually The Strangers.
Curious about humanity and eager to see if they can be used as part of their experiments, they casually manipulate events to see how humanity will react. And when it all starts going south, they take the most malleable subjects with them- along with the ever-mysterious Dr Schreber and as many examples of human technology and culture they can gather in time- and abandon the rest. The vision of Caleb and Abby running through the field towards a tree is just one last dream on their part before the Strangers erase their memories and begin experimenting.

The sheet of paper is an older version of the Death Note, and the Watchers are it's Shinigami
Come on, someone writes on paper and people die as dictated on the paper? Can't be a coincidence.

The Aliens are really the trans-human robots from A.I.: Artificial Intelligence.
After the death of David, the robots realize there are simply not enough specimens - robotic or human - to teach them about humanity.

Searching their research, they devote thousands of years to the idea of alternate universes, and find a way to cross over to a time-stream where humanity still exists. They succeed - only to discover that the sun in this universe is outputting massive flares that will eventually destroy humanity.

They know that Humans Are the Real Monsters from their own timeline; therefore they are unwilling to save all humans. However, they decide that saving a small population of children to seed a habitable world, will allow them to study humanity without the risk of being overrun by possibly robo-phobic humans.

Hence the actions that seem scary and horrifying to the humans - they have their own agenda and saving all of humanity is not one of them. And the "accidents" that occur within the 50 years of the movie is their way of "studying" humanity.

The kids that get shown the dates of the disasters are Seers of Doom.
Basically, everything is Sburb, minus the game element, and they're all on their way to becoming god tier.