Killstreaks are essentially where it counts your kills withim one life, and if you reach milestones (5, 10, 15, ect;.) It will display a message to the entire server (*Your Steam Name* Is on a Killing Spree, Unstoppable, Godlike ect;.)
You can collect a Killstreak Kit from Unboxing a Mann Co. Crate series No.#82, Which contains:
Rocket Launcher Killstreak Kit.
Scattergun Killstreak Kit.
Sniper Rifle Killstreak Kit.
Shotgun Killstreak Kit.
Ubersaw Killstreak Kit.
Gloves of Running Urgently Killstreak Kit.
Spy-cicle Killstreak Kit.
Axtinguisher Killstreak Kit.
You can also collect a random Killstreak Kit from completing a Mann Vs Machine Tour, As well as a Fabricator (Specialized or Professional Killstreak Fabricator)