So the short version is that the fibres don't cause mutation in DNA directly, so much as they damage cells in such a way that mutation becomes far more likely. DNA in a cell is contained primarily in the cell's nucleus. If the cell becomes damaged sufficiently to damage that, it can cause it to heal wrong. This can cause things like cell inflammation, which itself can damage cell DNA if it persists over a protracted period, and cause cancer over time that way. That's one element.

But another, more particular element is one of damaging our cancer prevention. IIRC, one of the things Asbestos does is damage immune cells particularly. Your body develops cells that might well progress toward cancer all the time. Tiny little growths of damaged cells appear semi-regularly in Human bodies, but are very quickly dealt with by the immune system, which hunts them down and destroys them before they grow much more than a few cells too large. It's why Immunology is proving to be a very promising cancer treatment for a lot of cancers, and it's also one of the reasons why immune-compromised people, such as people with HIV or AIDs, are at more risk of cancer as a secondary complication. And in the same way, Asbestos fibres can damage some of the immune cells whose job it is to destroy these early cancerous cells, preventing them from doing their job, and making cancer more likely.