What is the purpose of accrual period in survival analysis? Before reading the answer to this question, I, like the original poster, thought that we have to take the accrual period into account because some subjects might die before we start the treatment.
But, according to the answer, this isn't so. The treatment starts as soon as the subject is recruited. Consequently, we can follow the subject and assess the treatment from the very beginning. Why, then, is accrual period relevant at all? Why do I need to specify it when computing the power, e.g. using Hmisc::cpower in R?

The accrual period doesn't affect power (*with exceptions listed below). It affects when the hypothesis test will be fully powered.
For tests of hazard ratios, power is affected by number of events, not the total sample size. When you design a trial/experiment, you account for the difference between number of events and the total patients recruited. You assume some patients will not have an event during the observation period. So if you need 500 events for a hypothesis test with 80% power, you may recruit 1000 patients and end the trial when 500 events have been observed.
Let's say that if you recruit all patients at the same time you'll observe the required 500 events after 1 year. In reality, patients enter the trial staggered over the accrual period. Patients who joined the trial later will generally have events later. This doesn't* affect the power since patients will have an event at the same relative time in the trial, but the calendar time of the analysis at 500 events will be later.
*In some situations accrual can affect power, but this is due to the day-to-day realities of running a trial/experiment. The power calculation is unaffected by accrual. The Hmisc::cpower function (and the underlying theory) are adjusting for when events happen quickly relative to the accrual period. If one treatment performs poorly and events happen quickly relative to the accrual period, you might reach the required number of events before accrual period ends or before one treatment group is filled. It's also possible to recruit patients with a poor prognosis early in the trial which accelerates the time to reach events and dilutes the treatment effect.
