It's really a question of, 'why is a free market better than a restricted one?'. From a purely self-interested American's standpoint:
1. It's dangerous to have the government picking winners and losers; they can use that power arbitrarily to support or punish political allies/enemies, which creates corruption and economic inefficiency. That's happened recently with even popular, successful American businesses such as the NFL, Amazon, and, potentially, AT&T & Time Warner.
2. The U.S. has only a fraction of the world's resources; limiting access to the rest of the world's resources is like arbitrarily limiting potential investors in your startup to only those in your zip code. Now the U.S. company's investors will have to take the second best offer, if there is one, and the employees and customers also may pay a price. Again, imagine it was your startup and you found a foreign buyer, made a deal you loved - and then the government stepped in and stopped you.
3. Limiting access to those foreign resources limits U.S. business productivity, which raises prices and depresses wages. For example, limits on non-U.S. steel increased the cost of products that use steel and reduced productivity of companies that use it (such as the auto industry, the construction industry, etc.). Effectively, we all could be paying lower prices but instead we are subsidizing less competitive businesses (in this case, whoever buys MoneyGram) with a form of corporate welfare.
4. If the U.S. doesn't buy products and accept investment from other countries, they won't do the same for the U.S. The U.S. is only ~5% of the world's population and ~20% of global economic activity; the largest markets for many U.S. products are China and India (including smartphones, IIRC).
5. There's the fundamental issue of liberty: I don't take it to an extreme, but generally MoneyGram's owners should be free to do what they whether or not their neighbors and government officials like it. (Of course, there are many legitimate exceptions to that.)
But the whole premise is debatable. Why wouldn't we want someone from another country to invest, in MoneyGram or in your startup? What's the difference if they were born on one side of an invisible line in the ground or another? Is it better to have these arbitrary political and economic barriers? Why wouldn't we want someone in China to become wealthy?