Companies are always looking for new customers. Especially with the economy in trouble, companies need to be creative to find more people to sell to. Although there are some complications with international markets, it is now easier than ever to sell to anyone anywhere in the world though use of the internet and easy shipping and customs.
Firms seek markets (international or otherwise) in order to make money.
Firms might think that foreign markets are promising if the foreign country has a large population that might want the firm's products. That is why, for example, American car manufacturers would try to enter the Chinese market. They see that China's middle class is huge (numerically) and growing and they hope to sell cars to that class of people.
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The excessive profits and the poor development of the markets in the nearby area are the "good enough" reasons to make a company to seek international markets.
In their turn, international markets offer incents like fix assets, low wages, human capital, access to raw material, lower unit costs to attract capital infusions for their development.
International markets could offer fewer restrictive regulations that could create the premises for a business to grow. Anyway, it seems to be a "win-win situation".