How does argon-argon dating work - need a simple but not-wrong explanation
As noted in the comments the wikipedia articles (at the time this question was submitted) are contradictory. There are quite a few steps to the logic of how argon-argon dating works but none are too complicated, although I won't go into all of the possible interferences. One thing to keep in mind is that high-precision isotope measurements always measure ratios between isotopes, not absolute concentrations.

To understand argon-argon dating, you need to understand potassium-argon dating.

The isotope potassium-39 makes up about 93% of natural potassium. Potassium-40 is radioactive but has such a long half-life that it is primordial - it has been around since the earth was being formed. Because of this, we can assume that the potassium-40:potassium-39 ratio in minerals at any time is a constant. Potassium-40 decays in 2 different ways. Most decays by beta decay to calcium-40. About 10.7% decays to Argon-40 by emitting a positron. 

If the argon-40 stays trapped in the crystal and you can measure the ratio of potassium-40 to argon-40, then you know how long it has been since the mineral formed. This also assumes that there is no other source of argon like trapped air. A problem is that it takes two separate measurements to get the potassium concentration and the argon isotope ratios, increasing the uncertainty. Argon-argon dating gets around many of the issues by measuring only multiple isotopes of argon.

The trick is to irradiate the sample with neutrons along with samples of known age. Some of the potassium-39 forms argon-39 by an n,p reaction. By converting potassium-39 to argon-39 then measuring the argon-39:argon-40 ratio, you can calculate the sample's potassium-40:argon-40 ratio, remembering potassium-40:potassium-39 is fixed. The standards of known age are used to account for differences in the neutron flux during irradiation. Argon-36 in the sample accounts for any air contamination.

Argon-argon dating, ideally assumes that:


all argon-40 in the sample comes from either atmospheric argon or from decay of potassium-40 in the sample
all argon-39 in the sample comes from irradiation of potassium-39 in the sample and as noted above, if you know potassium-39:argon-40 ratio then you know the potassium-40:argon-40 ratio because the potassium-40:potassium-39 is constant (at this point in time)
all argon-36 is the sample is from atmospheric argon


The actual calculations are pretty simple algebra, combining all the pieces.