Why do they build astronomical observatories in the southern hemisphere?

really? well for starters, the more observatories there are the astronomers can work at once, it's not like there is only one guy looking into space going hey i found something guys, and lots of other countries like to develop there own astronomic studies groups, mostly formed out of their universities, and finally you can't see through the earth when you are trying to see the polar opposite of the location of your observatory

We put a lot of observatories in Chile because it's very dry and isolated in the mountains - you don't have many problems with light pollution or weather, and you're up high above a lot of atmosphere.  It's not necessarily because it's in the southern hemisphere.  The close we are the equator the better; we can see a larger percentage of the sky.

Antarctica is probably the best viewing location on Earth because the air contains no moisture.  That means no clouds, and clearer skies.  During its winter it also gets 4 months of constant night.  The southern skies are also more interesting in some ways, since it contains both Magellanic Clouds.  There is a robotic observatory somewhere on the inner plateaus.

One reason is coverage. There are plenty of sites in the Northern hemisphere, but few in the Soutern. Sure, you can have one big observatory in the Southern hemisphere and place it high enough so it doesn't get too affected by weather. But you may still get some cloudy nights, and some types of observations do require regular monitoring so if you want to observe on a given place and it's cloudy there, it's as if there wasn't any telescope there, you're dependent on other sites having clear skies.

Another reason is that there are more sites with potentially clear skies in terms of light pollution, since the majority of the world population lives in the Northern hemisphere. Sure, there is also less landmass.

The third is seeing conditions. It's better to observe overhead than near the horizon, because near the horizon the light has to transverse a bigger portion of the atmosphere, therefore it gets more affected by turbulence and absorption. Overhead there's less atmosphere for light to transverse, so the images appear more steady and less affected by the water vapor and dust in the atmosphere (which produces some reddening).

The sky over Antartica is particularly dry, because all the water that could be in the atmosphere precipitates as snow. Moist air coming from the sea loses, if it precipitates, loses its moisture as it crosses the landmass. So you have a very dry air deep inside the continent.

In the South Pole, a different kind of telescope is being prepared as well, called a neutrino telescope. It's the "IceCube" experiment. Basically, they bury a set of "light detectors" called photomultipliers deep in the snow, covering a volume of about one cubic mile. The ice at those depths is more or less transparent. When neutrinos transverse the Earth, some of them react with matter and produce electrons or muons with very high speeds. Some of the neutrino-matter interactions may even happen reasonably near the active volume, inside the ice. Those particles, which are charged, when crossing a transparent medium at a speed greater than the speed of light in that medium (in this case, ice), produce Cherenkov radiation, which is the blue glow you see in a nuclear reactor water well. Those flashes of blue light are then measured to reconstruct the passage of those secondary muons or electrons, and from those they are able to reconstruct the trajectory and energy of the original neutrino. These are significant to investigate high-energy phenomena, either from the Sun (solar neutrinos) or from extra-galactic origins (cosmic rays). So this is a different kind of telescope, because it measures a different kind of particle (not photons, but neutrinos). I hope this makes your head spin some more.

You can only see 50% of the night sky from the north pole.
75% from 45 degrees north, but not all of that in one night.
100% of the sky is visible from the equator, but once again, not all in one night.

The same format is followed for the southern latitudes, but the parts of the sky you can't see from say, 45 degrees north, are circumpolar from 45 degrees south.

The most ideal places for observatories are away from major cities (light pollution) and on mountaintops or in deserts (astronomical "seeing" comes into play...  less activity in the atmosphere, and less atmosphere to distort you views).