In a PCR reaction DNA polymerase is responsible for building the new strand of DNA. However because of the mechanism by which DNA polymerase works it cannot just build a new strand opposite the old, it can only build off of existing DNA.
Primers bind to the strand to give the polymerase something to extend. An old lecturer of mine used to give the analogy that primers are like lazy students who need a bit of encouragement, primers are like giving a student the first paragraph of an essay to get them going.