On Harry's final night at Number 4 Privet Drive, the Death Eaters were ready to ambush him when he left. The Order arrived to protect Harry on his journey to the safe-house. At that time, the Order was led by Alastor Moody, one of the most cautious wizards ever. Why did he or one of the other powerful Order members, like Kingsley, not check that the area was clear before they landed? If the Death Eaters were hidden with disillusionment charms, would Moody not have spotted them using his magic eye?
The Death Eaters probably did not Apparate
Here is Harry's impression of the Death Eater ambush:
And then, out of nowhere, out of nothing, they were surrounded. At least thirty hooded figures, suspended in midair, formed a vast circle in the midst of which the Order members had risen, oblivious.
Harry Potter and the Deathly Hallows
Note that Harry says that the Order members have "risen, oblivious" into the circle of Death Eaters. This seems to give the clear impression that there was a circle to rise into, especially given that the Order members are described as "oblivious." In this context, "out of nowhere, out of nothing" seems to be figurative: Harry believes that the Death Eaters have been lying in wait, not that they simply Apparated. This seems likely, since, had the Death Eaters Apparated, Harry would probably have heard the distinctive "crack" of Apparition, multiplied by thirty.
Further, it is unclear that the Death Eaters would know exactly where to Apparate to. They know the date and general time that the Order will depart, but not exactly how they will fly on their way to the safe houses (probably).
So how did they remain hidden?
Simple: They did indeed Apparate, but while the Order were in the house. They knew the time well enough for that. If they had flown in beforehand, the Order would have spotted them. If they had hidden nearby, Mad-Eye probably would have spotted them. One can certainly Apparate while on a broom: Mundungus did it.
Moody did not see the Death Eaters, nor did the rest of the Order, because they formed a "vast circle," and it was night. Moody certainly can see through solid objects, and in the dark at least at short distances, but at long distances I suspect darkness might still impair his vision. Certainly distance would.
I believe the DE were not hiding there (at least not in large numbers) in case Harry came out, and instead they just apparated there when it was time to attack
The book just says the DE just appeared and it could have been either way, but the important point to note is that it was not the intention of the order to fight them at Privet Drive. First of all, the Order did not know that the DE had found out the exact date and time of Harry's transfer, and even if they did, the Order were more going along the lines of distracting the DE by having multiple Harry's going in multiple directions
All in all, I suppose they did some basic check of the surrounding, but didnot go into much detail as fighting the DE there was not their strategy at the time