Think of pulsars, in this context, as faraway precise clocks that chirp out a regular sequence of pulses [1]. We don't get to pick where they are, or how far

Think of pulsars, in this context, as faraway precise clocks that chirp out a regular sequence of pulses [1]. We don't get to pick where they are, or how far away they are, but they're scattered all over the sky. A gravitational wave passing through our solar-system/region-of-space will cause a very well-prescribed variation in the properties of spacetime. It will cause chirps from pulsars from some parts of the sky to arrive a tiny bit early and chirps from other parts of the sky to arrive a little bit late. As the wave passes by, those "early" and "late" signals will oscillate, making a readily-discernable signal. The tricky bit, or the elegant bit, is that the signals accessible through this technique are very slow, in the nanohertz [2]. So, the experiment must watch signals from pulsars quasi-continuously for a long time, long enough to see multiple cycles of the oscillation. It was clear from the outset that the approach had great promise. It was also clear from the outset that the measurement would take decades. We're starting to come to the time where the results are beginning to become quite interesting. There are signals, but they haven't yet oscillated enough to separate oscillations from statistical variation. The pulsar-timing arrays are a wonderful example of the kinds of things one can do by thinking (and funding) long-term. [1] Yes, they're imperfect, but the imperfections can either be handled or their statistical variation correctly handled when estimating the measurement's uncertainty. [2] If you just turned 31, you're about to turn one billion seconds old. Put it on the calendar!