Why should countries specialize in producing goods with which they have a comparative advantage rather than an absolute advantage?

You get a greater gain from trade when the trading partners produce goods for which they each have comparative advantage. This is true even if only one partner has the absolute advantage with every good produced.

The reason the gains from trade are higher under comparative advantage is that all trading partners are producing what they make most efficiently, generating more total output than would be generated under any other choice of who produces which products

A country might have an absolute advantage in producing a good because it has a large population, this doesn't mean that the country is more efficient than other countries. If a country has a comparative advantage, its opportunity cost is lower. Therefore by specializing and trading a good that a country has a comparative advantage with, the country can get more out of what it produces.