For starters, this phenomenon can be intuitively explained in simple words only by using basic electrical concepts.
First of all, we must specify that here we are talking about the ratio of voltage changes to current changes known as "differential output resistance" - Rout = dVout/dIout. Then, we can observe two cases:
If a voltage-type negative feedback is applied, the output voltage will stay constant when the output current varies (the op-amp output will behave as a voltage source). So dVout -> 0 and Rout -> 0.
If a current-type negative feedback is applied, the output current will stay constant when the output voltage varies (the op-amp output will behave as a current source). So dIout -> 0 and Rout -> infinity.
So the conclusion is: Voltage-type negative feedback decreases the differential output resistance while the current-type negative feedback increases it.
To better understand this phenomenon, let's examine the op-amp behavior in the ubiquitous circuit of a voltage follower in three typical situations. Tip: you can get a good intuitive notion about the mechanism of the negative feedback, if you think of the op-amp not as a fast amplifier but as a slow-acting device (like an integrator); this will allow you to get inside its behavior.
1. Undisturbed follower. To make a voltage follower with negative feedback, we just connect the op-amp output to its inverting input - Fig. 1. Thus we make the op-amp keep its output voltage equal to the input voltage. For this purpose, it "observes" the voltage difference between its inputs and changes its output voltage until it makes this difference (almost) equal to zero. The op-amp does this effortlessly because there is no disturbance.
Fig. 1. Undisturbed op-amp follower.
2. Rout "disturbed" follower. Now imagine that Rout appears. To emulate it, connect an external resistor with resustance Rout in series to the op-amp output - Fig. 2. But as there is no load connected (open circuit), no current flows through Rout. There is no voltage drop across it and the op-amp does not react to this intervention. The op-amp output voltage VOA and the follower output voltage Vout are the same. As above, the op-amp does this work effortlessly because practically there is no disturbance.
Fig. 2. Op-amp follower with output resistance Rout
3. Rout-RL disturbed follower. Now let's connect a load RL - Fig. 3. As a result, load current IL begins flowing through Rout and a voltage drop VRout appears across Rout. This drop is subtracted from the op-amp output voltage VOA and the follower output voltage Vout decreases. Since the op-amp "observes" this voltage, it begins increasing its output voltage VOA to compensate VRout. As a result, VOA = (Rout + RL)/RL and Vout = Vin. So the Vout voltage change is suppressed. The follower output behaves as a constant voltage source with (almost) zero differential resistance.
Rout and RL actually form a voltage divider (the "beta" in the feedback loop of the system).
Fig. 3. Op-amp follower disturbed by the output resistance Rout and the load RL
4. Non-inverting amplifier.
Fig. 4. A non-inverting amplifier presented as a disturbed follower
Very interesting... as though VOA is the amplified Vout (Vin)... and we can use VOA as an output (OUT2) of this "non-inverting amplifier". We have only to keep the resistances R1 (RL) and R2 (Rout) constant. So the conclusion is:
The non-inverting amplifier is a disturbed follower.
Note that besides the new "amplifying" output OUT2 the old "following" output OUT1 continues to exist... and we can use it as above.
This is not only an electric phenomenon observed in op-amp circuits with negative feedback. We can see it in many everyday situations where we overcome all kinds of obstacles to achieve our goals. In doing so, we turn from "followers" into "amplifiers."
Instead of listing many examples of this phenomenon, I suggest you try one of them right now. I will write an informal explanation of how the negative feedback reduces to zero the output resistance of the op amp. This will upset the mental equilibrium of those who think formally and conventionally... and they will react to this "disturbance" by trying to destroy it. The interesting thing here is that they will react to the explanation of this phenomenon by the help of the same phenomenon. Here is my "provocative" explanation:
In the disturbed follower (Fig. 3 above), the op-amp increases its output voltage Vout with additional voltage dVOA that is equal to the voltage drop VRout across Rout (VOA = VL + VRL = VL + dVOA). This additional voltage is proportional to the load current in the same way as the voltage drop across Rout is proportional to the load current - dVOA = VRout = IL.Rout. Thus the op-amp adds voltage VRout = IL.Rout while the output resistor subtracts the same voltage drop VRout = IL.Rout. So, my conclusion is:
In the circuits with negative feedback, the op-amp output acts as a "negative resistor" with resistance -Rout that compensates the positive output resistance Rout (since they are connected in series). As a result, the circuit has zero output resistance (Rout - Rout = 0).
This is an explanation in terms of resistances while the previous explanation above was in terms of voltages. Now we just have to wait to see the reaction to this "disturbance" (silence, -1s, negative comments, etc.)
We are ready to generalize our observations into a "philosophy". We can formulate, like H&H, "Golden Rules for Applying a Negative Feedback into Op-amp Circuits":
Close the negative feedback after the disturbance.
If you want a follower, take the output after the disturbance.
If you want an amplifier, take the output before the disturbance.
The disturbance in the examples above was proportional - Rout-RL (R2-R1) voltage divider.
(I suggest you visit two resources that illustrate the unique property of negative feedback circuits to compensate all kinds of disturbances. The first is a Wikibooks story based on a lab exercise conducted with my students in 2008. The second is an interactive Flash movie named Strange things can be put into the feedback loop. I created it in 2002 when I was highly impressed by the Tom Hayes's 'Student Manual for the Art of Electronics'. It was then that I first encountered a way of thinking like mine and I was very enthusiastic.)