Anti-mage is what is called a 'flash farm' carry.
His main asset is his 'blink' ability which combined with his slightly above average auto attack item efficiency due to lower base attack time and a battlefury (which allows him to quickly kill neutral monsters and more importantly minimize his time spent traveling between places to kill monsters [lanes, jungle, ancients] ).
In a competitive environment, normally what antimage's team tries to do is not lose too many towers/die too much and stall for him to buy his battlefury. After he has his battlefury he uses blink on every cooldown when not beating things to death and quickly acquires his manta style and then some combination of black king bar, abyssal blade, butterfly, heart.
Flash farm carries are less powerful when fully equipped than true hyper carries because they have abilities that allow them to acquire money faster but aren't actual combat abilities (for example, Alchemists' Goblin Greed is useless when there is nothing left to buy, Alchemist is also no longer run as a full carry but we're getting off topic now).
For example, antimage's flat damage bonus from his mana burn is much less useful than a damage multiplier like Juggernaut's critical strike. However it's pretty useful against Medusa who relies on mana shield (do not pick Medusa against anti-mage unless you are extremely confident that your Medusa can stall the game 20+ minutes after the anti-mage's team is done stalling the game).
AM is weak against push, especially early since all he does is hit people at melee range and he can't fight until he has battlefury and one more item. He is out carried by true carries if the game goes on long enough like Faceless Void, Phantom Lancer, Spectre.
SS on the other hand is this mostly a pusher who requires levels (level 6 ASAP is crucial) and a lot of down time because his spells are hideously mana inefficient. He sort of fixes AM's problem of being weak to push and needing space to blink around and get rich by dropping his wards at towers and trading towers (since AM is more mobile than every other carry, trading loss of map control is almost always good for him).
Countering SS requires forcing fights so that he wastes his ultimate outside of tower range. His wards do not get any stronger and become incredibly weak as soon as enemy heroes can one hit kill them.
Antimage has no disables and can't do anything if Rhasta is stun locked into death.
Dragon Knight while boring is almost always a good choice and simply a nightmare to fight against by un co ordinated teams. He is incredibly bulky, does respectable damage with his spells and auto attack, crushes towers with his poison and has one of the most terrifyingly long (and undodgeable!) stuns in the game. A DK with a Halberd (4k gold) is untouchable by an AM unless the AM has both a BKB and an MKB (8 or 9k gold).
Storm is fine but you have to buy an orchid and play more to get experienced enough to not overlap the pull and the silence and not run out of mana. Magic wand is a god send against antimages and most heroes in general. Also always jump on him from fog/out of range/invisible/etc.
Hard carry siren with an early radiance and godly Starcraft 2 level macro to farm even more than antimage is a viable counter especially in bot games as well since bots tend to be somewhat disorganized and bad at ending the game. It looks like this and is the most disgusting thing in the world.