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. 

The second point is to debunk the common answer given, which is "because DC won't go long distances". Sure it will. In fact DC is sometimes better for long distance (because you don't have capacitive or EM radiation losses). 

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. So the longer the distance, the more important it is to cut the line losses to a minimum. 
 
2. The primary way that power lines lose power is in resistive losses. They are not perfect conductors (their resistance is non-zero), so a little of the power that goes through them is lost to heat - just as in an electric heater, only there, of course, heat is what we want! Now, the more power is being carried, the more is lost. For a given amount of power being transferred, the resistive loss 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, pronounced "eye-squared-arr", "I" being the usual symbol for current in electrical work.) So you want to keep the current as low as possible. Low current has another advantage: you can use thinner wires. 
 
3. So, if you're keeping the current low, then for the same amount of power delivered, you'll want the voltage high (power in watts = EMF in volts multiplied by current in amps). e.g. to halve the current, you'll need to double your voltage. But 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 make sense for the long distance transmission lines. (that could be several _hundred thousand_ volts!) Practical generators can't put out extremely high voltages either (they would arc horrendously). So you need an easy way to convert from one voltage to another. 

5. And that's most easily done with AC and transformers. Transformers can be amazingly efficient: power distribution transformers routinely hit 98 or 99 percent efficiency, far higher than any mechanical machine. 

By contrast, to convert DC voltages you essentially have to convert to AC, use a transformer, and then convert back to DC. The DC-to-AC step, in particular, will have losses. Modern semiconductors have made this a lot better in recent years, but it still  generally isn't worth doing until you're talking about _very_ long transmission lines, where the advantages of DC outweigh the conversion losses. 

Another reason that AC prevailed over Edison's DC was that the AC system scaled better, as it permitted a small number of power plants far from the city, instead of a large number of small plants about a mile apart. Edison didn't just want to sell light bulbs; he (or, rather, his investors) wanted to sell lighting systems to businesses. There was no power distribution network and he didn't want to have to build one before selling light bulbs. At first he was selling lighting systems to commercial buildings, maybe some large apartment buildings; each building would have its own independent generator in the basement, just as you typically have water heaters today.  He was initially successful because he (unlike other developers of light bulbs) was selling and installing complete systems, generator and switchgear and wiring and all, not just bulbs. 

This would have saved a lot of the clutter of overhead wires in cities, but it was clear that this would not work well for small businesses or homes (what homeowner or shopkeeper wants to worry about keeping a generator running?). Westinghouse wanted to build a hydroelectric power generation plant at Niagara Falls - one plant to run all of New York City and beyond. Tesla designed an entire AC distribution system  involving 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. And, of course, AC works for light bulbs as well as for motors. 

Speaking of that... Yet another reason for preferring AC is that AC, and particularly the three-phase AC that Westinghouse's system used (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 (which interfere with radio), etc. Whereas an AC induction motor needs no commutator nor even slip rings. AC power transmission systems start with three-phase AC generators and maintain three-phase right up to the pole transformer. So they 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 AC power distribution has another advantage in not needing a dedicated "return" wire. (Just FYI, the system Tesla originally designed for Westinghouse was two-phase. They changed to three-phase after the work of Mikhail Dolivo-Dobrovolsky in 1888-1891.)

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 at power line frequencies will cause involuntary muscle contractions - paralysis - and heart fibrillation at far lower current than DC (about a tenth). [(See allaboutcircuits.com)][2] However the end-user connectors were designed to minimize risk of contact with live parts, and we keep making them better in that regard. 

(Aside: I have long held that the electrical transformer should be regarded as one of the basic machines, along with the lever, the inclined plane, the block and tackle, etc. They have the same property of trading off one thing for another. In the case of the mechanical basic machines it's power traded for distance, for an equivalent amount of work done; in the transformer it's voltage for current, at equivalent power. Hydraulic cylinder master-slave pairs should be in the "simple machines" list too. ;) )


  [1]: https://en.wikipedia.org/wiki/High-voltage_direct_current
  [2]: https://www.allaboutcircuits.com/textbook/direct-current/chpt-3/ohms-law-again/