This is a good question. There are several reason why nations trade with one another. First, no nation is self-sufficient. This is to say that all nations need the resources of other nations. For example, some nations have little in terms of natural resources. If this is the case, then they need to trade with other nations. On the flip-side, some nations have a plethora of natural resources, and so they want to sell them to other nations to make a profit.
A concrete example will be helpful. South Korea does not have oil, but they have technology to refine oil. So, South Korea has built partnerships with Middle Eastern countries to share their technology in exchange for oil. This is a trade that mutually benefit each other.
Another reason for trade is, because it make sense economically. In other words, certain good can be make much more efficiently and cheaply in certain nations. For this reason, it make sense to ship these goods overseas to make a profit. And it makes sense for nations to buy from them. This fact in turn creates further trade between nations.