Many critics believe that director Don Siegel's Invasion of the Body Snatchers (1956) has strong undertones that purposefully play off the national fears of Communism and McCarthyism. Though Siegel never denied this, he claimed it was inevitable that people would perceive the movie in that way. In reality, he claimed it was less about politics and focused on the universal feelings of identity and fear of dehumanization ("Invasion," Turner Classic Movies).
The paranoia created in Invasion of the Body Snatchers can really be interpreted in any way you can imagine. I think that is the point; that those pods can be any incarnation of your worst fear, whatever that may be. Some other interpretations are fear of spreading diseases (such as AIDS), nature overcoming our industrial societies, or our mechanized societies overtaking ourselves. But the thought that an alien plague could take over all of mankind in a matter of weeks is frightening to the core ("Invasion," Gadfly). Space and aliens and whatever else is floating around out there in the universe are truly the scariest ideas left in fiction because they are "the unknown."
What makes the film so eerie is the fact that it is not unusual. It takes place in a regular town, there are very minimal special effects, and no one seems to notice anything out of the ordinary. It definitely plays on our paranoia that anything could happen at any moment. Movies about serial killers, plagues, monsters and zombies--they all stem from the very basic sci-fi elements of fear and paranoia.
The "self" especially is a very sensitive subject. The self is made up of our emotions, our souls, the pieces of us that are considered characteristically human. Ideas like possession and cloning--both sort of touched upon in Invasion of the Body Snatchers--prey upon that fear of (literal) identity theft. The movie is universally understood as threatening, no matter how you perceive that.
"Invasion of the Body Snatchers (1956)." Turner Classic Movies 1 May 2006 (http://www.turnerclassicmovies.com/thismonth/article/?cid=89417&mainArticleId=89482).