Functional principle of comparator based rectifier
That's a lot more complicated than the typical precision rectifier I've seen.

Let's get the easy stuff out of the way.  R18 and C6 low pass filter the result out of the rectifier, then IC2 amplifies it.  Depending on the time constant relative to the intended AC frequency, this yields the DC "average".

At first glance D5-D8 might look like a full wave bridge, but it's really not.  It seems it's really two diode pass gates.  Note that the raw input is connected to the left corner.  Think of D5-D6 separately, and D7-D8 separately.

Look just at D5 and D6.  They block the incoming signal when the bottom of R15 is high, and pass it when low.  Each of the two diodes ideally has the same voltage drop, so when R15 is driven high, the input voltage appears at the output.

D7 and D8 work similarly, except that a positive voltage (on R16) is needed to pass the signal from input to output.

The mess of stuff between each of IC1's outputs and the diode gate control signals seems to be drive circuitry with a bunch of fancy feed-forward speedup compensation, although I'm not really sure about that.  I don't have time to analyze that carefully right now.

It can be a lot easier to come up with a circuit from specs, than to look at someone else's circuit to infer the specs it was designed to.  At least in this case you can assume competent design.  Linear app notes and sample circuits are usually quite carefully done.

The input circuit to IC1 is essentially a differentiator.  The comparator output will go high on rising slope of the input, and low on falling slope.  I haven't worked out how switching the pass gates on the slope of the input achieves the desired result, but I have to go now and get some real work done.

The first thing to do is look to see if Linear offers any explanation of how this circuit is supposed to work.  Sometimes they explain carefully, sometimes not.  Maybe someone else can jump in here with a more detailed analysis.