I'll state upfront that I'm not a biologist, but here's my attempt at the explanation.
The ELI5 version:
Think of the cell as a factory for proteins. The blueprints for all of the different kinds of proteins it can produce are encoded in DNA and stored in the foreman's office. When the cell wants to make a particular protein, the foreman copies the instructions from the blueprint onto slips of paper (mRNA), and sends them to the assembly line workers. This type of copying happens all the time, and is pretty reliable -- and even if it's not, the end result is just some wasted time and a junk part. The blueprints are unaffected. mRNA vaccines sneak some unauthorized instructions to the workers, who then run a ghost shift to produce the protein we want to cause an immune response to.
When we need to create a second factory, we need to photocopy all the blueprints for the new cell. This uses a totally different process (a photocopier assembled specifically for the job of copying whole blueprints onto more blueprint paper, and not the work-order slips sent to the assembly line). The photocopier is pretty reliable too, but occasionally it introduces artifacts, and eventually, enough of these errors can build up to create mistakes in the blueprints. The blueprints have instructions on how to deal with errors in a new factory (by blowing it up), but if enough of those failsafe instructions are obscured by copier artifacts, you now have a potentially cancerous cell.
The foreman's office doesn't usually make changes to the blueprints based on work order slips. It's possible with a special copy machine -- this is how HIV works, by sneaking that machine in -- but without a work order slip to blueprint copier, it doesn't matter how many fake work order slips we pass into the cell. They will only affect the production line, not the foreman's office.
The more detailed, and almost certainly wrong-in-the-details version:
There are three types of encoding for genetic material: in DNA, in RNA, and in proteins.
In order for the information encoded in genetic information to be active in a biological system, it has to be expressed as a protein -- a physical manifestation with binding sites and the ability to interact with other molecules. Ribosomes, a component of the cell, do the "assembly" part of transferring genetic information to a protein. This is called "translation".
DNA is the long-term storage of the information eventually expressed as proteins -- it's a collection of templates in long term storage. So there has to be a process to get information from the long-term storage of DNA, in the nucleus of the cell, to the ribosomes elsewhere in the cell.
Messenger RNA is the intermediary molecule which carries the genetic information from the DNA to the ribosomes. It's produced in the nucleus of the cell by a mechanism called "transcription", which copies a subset of a DNA molecule into a complementary RNA molecule encoding the same information. The DNA double-helix is "unzipped", a complementary RNA strand is assembled, and the double-helix reforms. The mRNA can then be transported to the ribosomes.
mRNA is a useful vaccine mechanism, because we're co-opting a late stage of the cell's protein production pipeline. We don't have to assemble the target protein by hand and figure out how to modify it make it stable for distribution -- a task that would need to be repeated for each protein we wanted to make. Instead, we can figure out how to synthesis mRNA, and we don't need to worry about either synthesis or stability for the end protein product.
When we talk about cancer and mis-copying, this is at a much earlier stage in the pipeline. When a cell splits into two cells through mitosis, we don't want a subset of the protein templates in the cell's genetic material; we want a complete copy. This is a separate biological pipeline, which involves the cell building up a mechanism to do the copying, dissolving the membrane of the nucleus, and ultimately producing and collating copies of all the DNA from the nucleus. This is an entirely separate process, and messenger RNA isn't involved at all.
There is a route for information to travel from RNA back to DNA, and this is the mechanism retroviruses use, and HIV is the most famous example. A complete retrovirus particle includes reverse transcriptase, an enzyme which allows them to transfer information from RNA back to DNA. This is the reverse of the normal DNA -> RNA pattern described above, hence the name _retro_virus. Without that enzyme or something like it, there isn't a pathway to convert RNA back to DNA, and human cells don't normally express an enzyme which has this function.