You might have heard about gene vs environment..With respect to your query, in short DNA methylation constitutes a major part of the environment. It is a natural process that occurs across the genome. If DNA methylation occurs abnormally(either hyper/hypo methylation) in the promoter region of critical genes involved in cell cycle. It will lead to changes in gene expression and thereby carcinogenesis will be triggered.
What is your specific query? From what I understand, you only want to know the role of DNA methylation in Carcinogenesis. So, I guess you know that DNA is continuously exposed to various environmental factors (reports state the amount to be about 100000 lesions per day in each cell) that may cause DNA damage, which are consistently repaired by the DNA repair mechanisms. Among them, one such factor is methylation across the genome. If this occurs persistently in the CpG islands of the some important genes and goes unrepaired, it leads to altered regulation of cell cycle leading to trigger of carcinogenesis.