Imagine a capacitor and a resistor connected in series (one form of an "RC" circuit). This is a Zobel network, I assume named after some dude named Zobel. Hey, I'm not an historian. It is placed in parallel with the speaker terminals (from + to -) and what it does is to remove the inductive impedance rise at high frequencies. All dynamic drivers have an inductive component due to the fact that a voice coil is exactly the shape of an inductor. This causes the electrical phase to become increasingly inductive at high frequencies, which can drive some amps into oscillation. Not normally a problem, but the Zobel network, if designed properly, will virtually flatten the impedance and phase at high frequencies and eliminate any possibility of the problem developing.
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Basically, it's an R/C circuit shunting the amp output to ground. One problem amplifiers have is the instability resulting from low frequency signals. The amps will oscillate at a high frequency when there is no inductive load at the speaker. The zobel network acts like an automobile shock absorber. The resistor is the spring, which absorbs the deformation energy and the capacitor is the shock piston which dampens the vibration. When an amp sees a capacitive load from high C cables or no high frequencies, it starts to oscillate, or the road's bumpy with the auto analogy and the zobel network matches the speaker load to a resistive load and dampens the isolation.
By removing the zobel network, you remove the matching network interface to the speaker that makes the speaker behave as a resistive load and high frequency inductive loads cannot be damped out. Yes, the amp can oscillate when there is a high inductive load from the speaker (high frequencies) but some crossovers have a zobel network which prevents a back-oscillation. If there is no ZN at the speaker, then no big deal - the ZN at the amp output will take care of it (unless the amp has an output inductor which could cause a problem, theoretically).
I think that the "outboard circuits" he provided you with are nothing more than a matching network consisting of an R1-R2-C1-(R1)C2 and an R3-L1-C3 circuit designed to make the crossover network appear resistive. If it works, more power to him, If it sounds better, then, hey, why not. Personally, I never seen a problem arising out of this and I think this is a non-issue as I trust amp designers to do the right thing when I plunks down my cash.
As said above , the impedance of a driver will show a peak at resonance, and then a slow rise from the nominal impedance with increasing frequency. A Zobel network typically consists of a series network of a capacitor and a resistor connected in parallel with the woofer. The purpose of the Zobel network is to cancel the rise in impedance which is caused by the driver's voice coil impedance. In effect, the voice coil impedance is resonated out using a capacitor.
Here are a couple short online articles that do a decent job with impedance graphs and pictures of a typical simple zobel circuit. Sometimes they help.