Overwatch – How to Solo Blood Moon Rising

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:

You can remove yourself from the channel typing /leavechannel general in the chat. If you are talking in general channel, you can just type /leavechannel

Right now, however, as soon as you restart Overwatch and sometimes when you leave a match you will rejoin the General chat channel automatically and will need to leave again.