There really isn't an ELI5 that covers any of the bases for this. So... It'll be ELI10-15.

Preemptive summary:

1. Humans have a lot of endurance. Way more than any other animal. We can chase down anything, given enough time. We win the marathon, not the sprint. And meat is a far more digestible food source than plant matter. But it doesn't hurt to still be able to eat those plants when we find something.
2. Humans have amazing communication skills, which allows for massive groups. A herd of intelligent apex predators that willingly work together.
3. Humans are smarter than anything around by huge leaps. Intelligence is a weird advantage, since only being a little bit more intelligent probably doesn't mean much. But being significantly smarter? That's kind of a big deal. And with our huge groups, we can accommodate that development time.
4. Mating is not random.
   We didn't "happen to" select for bigger brains, endurance, and strong communication skills. The bigger brains and endurance provided traits that were desirable and continuously selected for. Intelligence is kind of a weird "trait" in that it has a cascade effect on itself. The closest you get to other traits having that effect requires you to look at things like the dinosaurs and they were certainly successful with their exaggerated physical traits...it just didn't help them when all the various extinction events fell in on them.

How did humans become so smart?

Many reasons.

Physical Advantage:
Humans are endurance monsters. We don't win the footrace, we win the marathon. Every time. We are the best at long distance races in the animal kingdom. Seriously. This means we had a meat supply of which we could consistently take advantage. Meat is an energy and nutrition-rich food source that is easy to digest. Cows, etc., are basically all stomach for a reason. Plant matter takes SO much energy to process to get even the tiniest returns. Meat is the better option but obviously it is much more difficult to acquire.
Australopithecus, a precursor in our evolutionary tree, did not have the running capabilities we do today (the Achilles tendon being a key factor). The transition to the Homo genus 2 million years ago is marked by the appearance of things like the Achilles tendon (and other strong tendons). Tendons aren't the only factor that feed into the "marathon" advantage. Sweating. Our muscle fiber preference. Our bone and joint structure. Everything feeds into it and makes us inescapable hunters.

Social Advantage:
Animals that work in groups tend to be incredibly effective and widespread. Lions. Wolves. Both are widespread apex predators in their respective environments. A large group provides many benefits. One, it keeps you protected from the OTHER predators--it's much more difficult to pull a meal from a horde of angry animals than it is to catch one that's by itself. Two, infant development can be drawn out for much longer, allowing for better adults. A foal can run almost immediately after it's born. How much smarter does it become in the next 5 years? You have a cadre of adults that work together to raise the children, teach the children, and protect one another. And as a bonus, if one human fails, another might succeed and the group compensates for that on a daily basis. The human that was failing might be the successful one tomorrow and so on. The tiger by itself that fails in its hunt every day doesn't live very long and it becomes increasingly less effective until that point.
Additionally? Language. Think about how much your communication helps you in your day-to-day life. Strong communication skills mean more organization and planning. More organization and planning means more successful food acquisition. And the huge factor? Strong language skills allow for much, much larger groups. And if you want to think of it in another way: There are HERDS of intelligent apex predators.

Mental Advantage:
Average humans are smarter than the average bear. Well, why wouldn't an average human that's smarter than an above-average bear be better? And how about an average human that's smarter than a (relatively) genius bear? Well suddenly you don't need teeth, claws, and sheer brute force to successfully find food. You use alternative methods to acquire it that the (relatively) genius bear could never begin to imagine. And that's just what our children are capable of accomplishing, let alone an adult.

And in the end? Mating is not random. Keep selecting for bigger brains, better communication, and higher levels of endurance and that's what you get.

Why are we the only ones?

Our specific species? Because the other versions died out. We killed many. We integrated some (Eurasian ancestry means you have Neanderthal DNA). The measures of how much varies but as our species spread, it dominated.