Speed And Movement
The best advantage for the scout. Being faster than everyone else means you can have the whole map to yourself, if you can dodge the sentry guns and snipers. The most obvious use of this is to capture the enemies flag.
You may also come across some pipebombs set by a demoman has an area covered with pipebombs. There are two ways to possibly avoid getting killed:

1. Run towards them and cut back at the last second. The idea is that the demoman is likely to try and predict your movement as the scout is so fast and if they didn't they wouldn't be able to catch you. So, the demoman detonates their pipes when you dodge back, leaving you to run clear through.

2. If the pipes are not spread wide and concentrated in a small area, it is usually possible to run and jump past them. It is best not to jump straight over them, but through them instead aiming to land quite far past the centre of the explosion.

Weapons
For most classes their weaponry is their main advantage. Not so with the scout and it comes second. Equipped with only the axe, shotgun and nailgun, plus a low ammo carrying capacity means that many believe the scout useless for anything less than running from base-to-base. However, a half decent strafer should be able to take out any other class with the nailgun.
Be careful with your nails, 200 isn't alot at the rate they come out. Of course don't end up not using them at all, and it can be worthwhile spreading them between your enemies heads and other limbs.
Should you use up all of your nails, don't hang about to see if you can get anywhere using your shotgun. With just the one barrel you really are better off running for it. Same for the axe, obviously. (But always look out for discarded ammo)

The scouts grenades aren't exactly gonna keep anyone awake at night, but they can be useful. The caltrops can easily be used to spam an enemy respawn area, sorry I mean strategicly concentrate on. Dont try and throw the canister though, they simply fall around you so don't stay still. The best use is when your being chased, say down a tunnel. Press and hold any key you have bound to them and just keep running. They will drop and lie in your wake, ready for any foolish HW Guy to stand on.

The concussion grenade can be used to move enemies out of your way, for example when blocking the entry/exit of a tunnel/ramp. Just be careful not to hit yourself and apologise when you hit your team-mates.
 
Strafing
The effectiveness of strafing is enhanced even more with the scout due to its speed. The best method is the "diamond" or "circle" strafe. Basically this involves on concentrating on a certain point, and then strafing in a circular formation.