When is Overwatch set

From an interview with Michael Chu, a Senior Game Designer on Overwatch (emphasis mine):

If you look at a game like World Of Warcraft, the Horde/Alliance split is such a big deal and such a defining aspect of the game. But Overwatch, pretty much from the start, we didn’t want to restrict ourselves that way. Because the game is so much about having the flexibility to play all these heroes and finding all the ways they play well together or against each other. Because of that we decided that the gameplay of the Overwatch universe wouldn’t be bound by that.

Our story justification – what I usually tell people – is that I think of it as, it’s clearly not canonical, but I think of it as how in comic book stories sometimes Iron Man and Captain America fight each other. Maybe in this instance on this map they’re fighting – for some reason they have some ideological difference or something that’s bringing them against each other.

But I think ultimately it was important for the game that you were able to play whatever characters you want. Blizzard has this development philosophy which is ‘Gameplay first’ and this is definitely an example of where we’ve gone with that.

What you are looking for is the Custom Game option in the Play menu. You can choose the map, the map rotation, the difficulty and amount of AI players, etc. This should allow you to play privately or with friends in order to increase your knowledge of maps and heroes.

http://overwatch.wikia.com/wiki/Custom_Game

Edit: As to how the youtubers and others fly around the map, they use the custom game menu, and when viewing the team lineups page, they right click on their name and swap teams to the Spectator team. Then they start the match. This video shows how it can be done at around the 10:00 mark: