An ESR meter won't tell you the in-circuit capacitance of a device, just its series resistance. It can be useful if you know how to use it and know what the readings mean.
There are some caveats--you don't use one on a powered circuit and if there are several capacitors hooked in parallel (common on computer PSUs), it can only measure the composite ESR, not that of any particular cap.
Basically, a low-voltage (so as not to involve semiconductor bandgap) AC signal is applied and the current flow is measured. All the other stuff is refinement and window-dressing. If you're of a mind to "roll your own",
here's a popular DIY ESR meter project.
The whole theory hinges on the fact that wet electrolytics dry out with age, which causes the effective internal resistance to rise, which also (remember I2R?) causes self-heating to increase and accelerate aging.