What exactly is the argument for Free Trade?
The theory behind free trade is denominated comparative advantage, and as mike stated, the advantage is that a country, as a whole, benefits from specializing in the fields at which it is best, as it will get more than it would if it would produce everything alone.

In practice, a big advantage is the ability to operate for a bigger market, reaping economies of scale. This is particularly advantageous for small countries.

The downside is that, although free trade will make the country as a whole richer, it wont make the society necessarily equally richer, so it wont guarantee that an economy will develop. This connects with the concept of exploitative trade and fair trade: by entering international trade, developing countries might leave their economies vulnerable to foreign companies or governments, sometimes in critical sectors such as food and energy.