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Joined: Oct 2009
So there are 4 ancient era units (besides warriors). Axes, spears, archers, and chariots.
Axes and chariots only require techs you want ASAP anyway. Spears also require hunting, which often sucks, and archers require that and archery too. Ease of building them assuming you have the necessary resources therefore goes something like axe > chariot > spear > archer. This is incidentally also the order of their effectiveness in countering barbs. You really just want to build axes (highest base strength and can attack barb warriors in any terrain) and/or chariots (high mobility but actual chances of dying in barb combat).
War Chariots own both of those units. We can just build primarily one unit and it counters the heck out of everything that's not a spear. On top of that, it has great mobility and is cheaper than an axe. And no one wants to deal with them because that means researching hunting and building a stupid unit that is mediocre vs barbarians. Not to mention if we find someone without metal (due to slow BW and/or not prioritizing settling it and/or not getting hunting before we can disconnect it) we have a huge advantage over them instead of only a small advantage. (Normal chariots don't get odds against archers in cities, nor of course against other chariots.)
Btw here's a comparison to the best military UU there is:
Cataphract vs normal Knight is +20% strength, loses immunity to first strikes.
War Chariot vs normal Chariot is +25% strength, gains immunity to first strikes.
Now, chariots are certainly a bit more easily countered than knights (pikemen don't have 10 strength!) but WCs get a much bigger boost than you might think just looking at the 1 strength difference. They're great.
Oh and as has been pointed out, it's a good threat to have handy for diplomacy.