I don't know how strong your biology, specifically genetics, background is, but they do explain how they established the rate of change (the second paragraph under the heading "The slowly evolving coelacanth"). I will try to keep my explanation of that second paragraph succinct and simplified, so it might not satisfy your curiosity. But, you can also consult the methods, which are in a linked PDF document (bottom of page 8, through to page 10).
Due to random copying errors during DNA replication, a base pair substitution can occur in a gene. The rate at which these substitutions occurs in a population is relatively fixed, and low. These mutations accumulate over time in a population. If you compare the sequence differences between two species you can estimate how long ago they diverged based on the mutation rate.
The authors compared the degree of difference between the coelacanth genome and the genomes of three cartilaginous fishes. They did the same for the lungfish, chicken, and mammalian genomes. Now look at Figure 1. The ancestor of the coelacanth, lungfish, chicken, and mammals diverged from the ancestor of the cartilaginous fishes at the same time. Therefore, if the coelacanth et al. genomes evolved at the same rate, the degree of difference from the genomes of the cartilaginous fishes should be the same. It is not: the rate of divergence is lower for the coelacanth.