One of the reasons is that far more Muslims die due to Muslim terrorism than do non-Muslims; groups like ISIS make the Westboro Baptists look inclusive.
But given that terrorism doesn't require much support even a small number of Muslims who believe that since Muhammad spread Islam via violence it's ok to use violence in the name of Islam can create a disproportionate amount of damage.
Further many Muslims believe that Islam should be in charge of civil affairs, i.e. sharia law, even if those Muslims don't support violence.
That creates a problem in the US. As a Catholic I look to my faith to define my morals and then I try and implement those morals through the law following the legal processes. But since the Church does not say we should be intolerant of other faiths or that the Church should run civil society my beliefs don't conflict with the Constitution.
Many Muslims however believe that the Quran comes first and that the Constitution is a bad thing; since Islam does not teach religious tolerance in general. That means that those Muslims, however many or few they may be, can't really become Americans; they can't assimilate.
Since America does not benefit from bringing in people who reject America's founding principle it would seem obvious that screening would be immigrants would make sense.