The first point to make is: We don't always use AC. There is such a thing as [high voltage DC][1] for long-distance power transmission. However its use was rare until the last few decades, when relatively efficient DC-to-AC conversion techniques were developed. 

But, yes, AC has been used traditionally. The "why" is because of a series of "a leads to b leads to c leads to...":

1. You want to lose as little power as possible in your transmission lines. And all else being equal, the longer the distance, the more power you'll lose.
 
2. For a given amount of power being transferred, the resistive losses in the transmission line is proportional to the _square_ of the current! (This is because power (in watts) dissipated in a resistance is equal to current in amperes, squared, multiplied by the resistance in ohms. These losses are commonly called "I-squared-R" losses, "I" being the usual symbol for current in electrical work.) So you want to keep the current as low as possible. 
 
3. So, if you're keeping the current low, then for the same amount of power delivered, you'll want the voltage high. To halve the current, you'll need to double your voltage. But if you double your voltage, this will cut your losses to one fourth of what they were! That's a win. Now high voltage does have its issues. The higher the voltage, the harder it is to protect against accidental contact, short circuits, etc. This means higher towers, wider spacing between conductors, etc. So you can't use the highest possible voltage everywhere; it isn't economical. But in general the longer the transmission line, the higher the voltage that makes sense. 

4. Unfortunately you can't deliver power to the end use point (wall outlets and light sockets) at the high voltages that are desirable for the long distance transmission lines. This means you need an easy way to convert from one voltage to another. 

5. And that's most easily done with AC and transformers. To convert DC voltages you essentially have to convert to AC, use a transformer, and then convert back to DC. 

Another reason that Tesla's AC scheme prevailed over Edison's DC was that Tesla's AC system scaled better. Edison was basically selling lighting systems to businesses and he envisioned that each commercial building would have its own generator in the basement. He was initially successful because he (unlike other developers of light bulbs) was selling complete systems, generator and all, not just bulbs. Tesla saw that this would not work well for small businesses or homes and thought in terms of a central power generation plant. He designed an entire AC distribution system involving three-phase AC induction generators, step-up transformers to boost their output as necessary for long distances, then conversion through a series of step-downs to what is called "distribution voltage" and then finally to the lines that are connected to houses and light commercial buildings. This was a far more scalable system than Edison's.

Yet another reason for preferring AC is that AC, and particularly the three-phase AC that Tesla's system uses (everywhere except at the last drop, from pole distribution transformer to house), was and remains far better for running high-power motors. All practical motors are really AC motors at heart; "DC" motors use commutators to switch the polarity to the coils back and forth as needed, to maintain rotation - essentially they make their own AC internally. But commutators require brushes, which wear out and require maintenance, they make sparks, etc. Whereas an AC induction motor needs no commutator nor even slip rings. Tesla designed his power transmission system to start with three-phase AC generators and maintain three-phase right up to the pole transformer. So it can easily deliver three-phase where it's needed (medium and larger commercial and industrial), but the pole transformer can tap off single-phase for homes and light commercial use. 

Three-phase has another advantage in not needing a dedicated "return" wire. 

During the "war of the currents" Edison made much of the greater danger of AC. It's true that a given level of current, through a given path through the body, is more dangerous at AC than at DC. That's because AC needs lower current to cause involuntary muscle contractions - paralysis. However the end-user connectors were designed to minimize risk of contact with live parts, and we keep making them better in that regard. 

  [1]: https://en.wikipedia.org/wiki/High-voltage_direct_current