The idea is that the coevolution of parasites and their hosts produces rapid environmental change for each one. The host is the environment for the parasite, and the parasite is the relevent environment for the host. For example, a host might evolve a new type of immune response to a particular parasite. If the parasite does not quickly evolve a way to overcome the defense, it will go extinct. The environment of each species deteriorates as its antagonist evolves a new adaptation.
Sexual reproduction is more likely to be advantageous if environments change rapidly.
And parasites and their hosts may themselves generate environmental change at a speed that makes sexual reproduction advantageous in the short term, i.e. every generation as opposed to physical environmental change every few hundred years or more. Sexual reproduction is therefore a means to avoiding extinction. In this case running is the metaphor for coevolutionary change. Under the Red Queen mode of evolution, the environment of each species deteriorates as its antagonist evolves new adaptations, and natural selection continually operates on host and parasite to keep up with adaptations in the other species. From the point of view of the most abundant lineages in the population, the environment continually tends to get worse.