We can't "stay out of" "their" countries.
First of all, there are no "their" countries. Yes there are countries in which a majority of the population is Muslim, but that doesn't make that country "owned by Muslims" or a "Muslim country". I reject that thinking. To see why I object to the phrasing: the majority of Americans are Christians but that doesn't mean it is valid to call it a "Christian country", or to say to a Jew, "why are you living in America, that's their [Christians'] country, not yours!"
Second, you seem to ignore/forget that people in some of those countries which are majority-Muslim, have entered into business, political or other kinds of relationships with Westerners.
For example, to develop oil fields, Western Company X signs a deal with "Muslim country" powerful-guy Y. Company X proceeds to make a large investment in moving equipment, manpower, etc to that country, to set up an oil extraction facility. This is property which belongs to Westerners. To say "we should stay out" is to say "we should let 'Muslim countries' steal the stuff we've already moved there".
Also, typically in such cases, much of the equipment/infrastructure is not capable of being operated/maintained by the natives anyway; thus to say "we should stay out" is to say "we should let those natural resources go un-extracted". But actually, a more likely result would be that some other Western country with less scruples, i.e. France, moves in and takes over operations (along with resulting profits). And we still get the hatred, of course.
There are other special cases worth mentioning, e.g. Egypt, which most people say we are not "staying out of", because of the fact that we send them money. But to say we should "stay out of" Egypt (=not send money) is to say that we should abandon the Egypt-Israel peace agreement (because that's the only reason for the money in the first place). (Ironically the example of Egypt is often raised by the very same people who idolize the guy that put us "into" Egypt in the first place, and supported giving him a Nobel Prize.)
2.Keep them out of your Country---or they will try to turn your country into theirs.
Luckily we have a separation of church and state, and a Bill of Rights protecting our basic liberties, which I believe will be effective in preventing that. European countries, alas, often do not seem to be faring so well.
Sounds like good advice to me.