The m in mRNA stands for messenger.
I'm failing to find anything I consider to be a succinct explanation in layman's terms (other than the one in the piece under discussion: all that the mRNA vaccines do is to add a dose their own specialized sequence that then makes your normal cell machinery generate that spike protein so that your body learns how to recognize it.).
But from what I gather, mRNA vaccines basically inject your body with a coded recipe for how to manufacture the immune stuff it normally manufactures in reaction to an infection, only without having to first be infected. Not even with a dead version of the virus, as some vaccines use for trying to prompt the body to figure this out more safely than by actually catching the infection.
A few sources I looked at, without feeling satisfied with their explanations:
https://en.wikipedia.org/wiki/Messenger_RNA
https://www.genome.gov/genetics-glossary/messenger-rna
https://www.nature.com/scitable/topicpage/translation-dna-to...
Edit: Thank you to everyone supplying a more accurate explanation in layman's terms. Much appreciated.