I seem to recognise that quote, it is from either Wikipedia or my web site, I wrote both.
Dunne's argument was that, in a four-dimensional block spacetime, such as Minkowski space, there was no unique moment of "now". On any given timeline, any "now" would have to be chosen arbitrarily. But clocks tick away steadily and we experience time as moving, the "now" moment travels along the timeline at the clock's steady pace. But how can one establish that the pace is steady, that clocks run true? (She had no clue and gave him a flea in his ear).
At the age of nine he had been convalescing and had challenged his nurse about this; did Time comprise the waystations of yesterday, today and tomorrow, or was it the travelling between these waystations?
He eventually concluded that one cannot use a thing to measure itself, one needs a ruler. So he could not use his timeline to measure the rate of passage of time, he needed a different ruler. His argument was that this ruler had to be a deeper level, another dimension, of Time.
This second time dimension was indeed subjective, as some comments have suggested. But Dunne regarded it as essentially real, and this led to the idea that consciousness existed in this second Time dimension, rather than the base physical Time of the brain.
One cannot say that his logic had any more basis than that. It promptly led him to the problem of measuring the passage of time in the second dimension, resolved in the same vein by yet further dimensions of time and consciousness, and so on into an infinite regress. He justified this solution with reference to a similar regress of time which (I think) McTaggart had been wrestling with.
But Dunne was in no way an academic philosopher and his relentless explanations and justifications, though claiming clarity, became notorious for their obscurity.