High voltages Transmission lines?
The goal of high voltage transmission is to minimize power losses.

What causes power loss?

The main enemy is Joule's first law, or \$I^2R\$ losses. The resistance of our transmission line causes a loss in power with square proportion to our current. 

How do we fix it?

Say we want to transmit some amount of power, we'll call it \$P\$.

We know \$P=VI\$. This means that 1 volt and 1 amp has the same amount of power in it as 1000 volts and 1 milliamp.

Now, to minimize our \$I^2R\$ losses, we need to minimize both \$R\$ and \$I\$.
\$R\$ is hard to change. We have 1000s of km if not more of wire we'd have to change. Also, \$R\$ is a direct relationship, whilst \$I\$ is in a square relationship, so changing \$I\$ has a bigger effect. What's easier to change for our AC power system is \$I\$. 

We do this with a transformer. An ideal transformer allows you to put in some amount of power at a given voltage, and receive the same amount of power out at a different voltage. 

Now because \$P=VI\$ that means it also changes our current. \$V\$ and \$I\$ are in an inverse relationship in power, if \$V\$ goes up, \$I\$ goes down.

Thus, we increase \$V\$ to decrease \$I\$. By making the voltage on the power line higher, we lower the current, which thereby lowers our lost power \$I^2R\$.

For a given power we want to transmit (\$P = VI\$), a higher voltage means less current, which thereby means if we use a higher voltage, we minimize currents, minimizing \$I^2R\$ losses.

Why don't we jack the voltage way up?

So we increase the voltage and decrease the loss, so why not crank the voltage as high as we can? It turns out that too high actually increases our losses, due to a different effect. Eventually, we reach a point where the air surrounding our wire starts to ionize.

Ionization causes air to stop acting like an insulator, and allows current to flow. This process is known as coronal discharge, and means eventually we start losing some power as there is a new path to ground or another transmission line present due to the high voltages. It also slowly degrades our wire as atoms are pulled off the surface of the wire.

If we went even further, we'd eventually hit electrical breakdown, which means we now have arcs flying through the air. Arcing is extremely bad, and will both degrade our wire rapidly, as well as give what is for all intents and purposes a short, causing rapid power loss.