Roughly: Protein descriptions are "read" from DNA into mRNA, which is then used by ribosomes to synthesize proteins. An mRNA vaccine adds additional mRNA with instructions to synthesize the vaccine-specific protein to the cells. So it uses some of the same mechanisms that are also used to generate proteins from DNA, but does not involve DNA itself.
Roughly: Protein descriptions are "read" from DNA into mRNA, which is then used by ribosomes to synthesize proteins. An mRNA vaccine adds additional mRNA with instructions to synthesize the vaccine-specific protein to the cells. So it uses some of the same mechanisms that are also used to generate proteins from DNA, but does not involve DNA itself.