High voltage electricity transport
The benefit of using high voltage is that we can deliver the same amount of power with lower current through the transmission lines. Lower current reduces the losses due the resistance of the lines. This is true whether we use AC or DC at high voltage (and, in fact, high voltage DC transmission is becoming more common)

The benefit of using AC is that, given the technology of the late 19th and early 20th century that was present when our transmission network was developed, it is much easier to convert high voltage AC to medium or low voltage AC for delivery to the end customer. We can do this using transformers. No comparably affordable and reliable technology was available for converting between DC voltages when the power network was designed and deployed.


  And if the voltage is higher how can the current be lower? 


Notice I said above "to deliver the same amount of power". 

A 100 W lightbulb in a 240 V country uses the same amount of power and produces the same amount of light as a 100 W lightbulb in a 120 V country. But in the 240 V country, the lightbulb is designed with higher resistance so that it draws less current than the 100 W lightbulb for use in the 120 V country.

Similarly, if we have 20 residential customers drawing 20 kW of power in aggregate, and we feed them with a 20 kV line (using a transformer to step that down to 240 or 120 V before delivering it to their homes), that line will carry less current than if we feed those customers with a 10 kV line.


  The only difference I can see, is the relief on the cables because of the polarity switch, is that why AC is better to transport electricity over long distances?


This is a bit off the main focus of your question, but actually AC isn't better than DC as far as the wire losses are concerned.

First, because the AC signal spends some of its time near 0 V, the peak voltage of the AC waveform must actually be higher to deliver the same power as a given DC voltage. For example, when we say we have a "120 V AC" power source, we mean the AC voltage has a root mean square (rms) voltage of 120 V, as this is able to deliver the same power to a resistive load as a 120 V DC source. But the peak voltage of this AC source is about 170 V. This means the wire must be insulated to prevent arcing at 170 V rather than just 120 V.

Second, because of the skin effect. This means that AC currents tend to mostly flow on the outer surface of a wire, while DC currents can flow through the full cross-section of the wire. The effect is small at the fairly low frequencies we use for power transmission, but it still means that the transmission wires have effectively higher resistance when carrying AC than when carrying DC.

So again, the main reason for choosing AC power transmission is to be able to use transformers to convert between voltages rather than because AC is inherently better.