An important line in this article: "The cell model that we used in this study is a carcinoma cell line, with active DNA replication which differs from non-dividing somatic cells. It has also been shown that Huh7 cells display significant different gene and protein expression including upregulated proteins involved in RNA metabolism." This is a huge caveat. The cells in the experiment are not similar to those in the human body, and the differences are very difficult to assess. While integration of RNA into the DNA is not impossible, it's something to be expected with a viral infection. So even if this indeed happens in human cells, the question is how large the risk is comparing to the risks of contracting covid.

Simple explanation: the researches found the RNA from vaccine integrates itself into the genome, hence DNA, and is being transcribed by the cell. This can alter protein structure, as in: make a mess and harm the cells. What they're describing is not unthinkable in viral infections but the research work here is not very good and the headline is amplified without much justification. Bottom line: don't panic, do vaccinate.

Resource: am a molecular biologist