Before I answer this question, I have a question of my own.... are we talking about Americans or the United States? The reason I am asking is... I live on the continent North America, but I am a citizen of the United States. To reference "Americans" that could mean Canada, Mexico, Ecuador.... etc.
Let me assume you are referring to the United States (cause I can't answer for any other country). We get involved because we, majority of the time, are asked to. The governments of problem areas call to us asking for our help in their issues.... why, because of our strong military. Do we want our people there.... HECK NO! But then we also think of the atrocities that occur in many of these areas and we reluctantly go. We either are the solution.... or worse then the problem because we turn our backs on them. So we go and "police" or train as best we can.
This has been occurring since the US entered the first World War and was found to be a very powerful military presence (truth is, we held out until the last second making agreements that made the country wealthy, then went in with fresh soldiers to defeat the enemies whose own reserves were nearly tapped. The we did it again in WWII). As a result, we have been named as a power house and called on in every crisis.
Or so that is my opinion.