What does it mean by “end effects can be ignored”
They are considering a model of a DNA sequence of quasi infinite size. They consider a case where the DNA sequence is so long that the fact that it has ends will only negligibly affect the conclusions of their model. This is likely the same as considering the DNA is circular.

To give you a similar logic, imagine a species where individuals live in different populations. All population can be placed on a one dimensional axis. The migration rate from one population to the neighbouring population is 0.1 (this is called a "stepping stone model" btw). Hence, the probability of not migrating is 0.8 (1 - 0.1 - 0.1). I can draw conclusions from such a model but it would be important to specify that I neglected the end effects and hence, I assumed there are a lot of populations (or that the population form a loop). Why did I neglect the end effects. The last population on each side cannot have a migration rate of 0.1 toward a population that does not exist. Hence, their migration rate must differ from the other populations. This is commonly called edge effects (rather than "end effects")