You're right, most of those Google hits don't really describe the mechanism of anode break
Anode break is an "old" term, from long before microelectrodes were invented.
Physiologists used extracellular electrodes to stimulate nerves. Stimulus electrodes were
applied to the outside of nerve trunks. An extracellular electrode is called a cathode or an
anode, depending on whether the stimulating apparatus (often just a battery in series with
an on/off switch) drives it - or +. It was observed that a cathodal stimulus would excite
the nerve when the stimulus current is turned on, but an anodal stimulus excited the nerve
when the stimulus current is turned off (hence the term "anode break").
A cathodal stimulus excites a nerve by reducing the electrical field across the
membrane of adjacent axons ("depolarizes" them). This opens sodium channels and
triggers a spike.
What accounts for anode break excitation? It helps to know that, at rest, potassium
conductance is slightly activated, and sodium conductance is partially inactivated.
An anodal stimulus increases the transmembrane electrical field of adjacent axons
("hyperpolarizes" them), and this has two important effects: it decreases resting
potassium conductance, and it relieves sodium channel inactivation. When the anodal
stimulus is turned off, membrane potential quickly springs back from its hyperpolarized
level. Since fewer sodium channels are inactivated, more of them open up, producing
an inward (depolarizing) current. Meanwhile, potassium conductance is slower, so it
lags behind. Consequently, membrane potential overshoots the resting level, making
even more sodium channels open, generating even more inward current and producing
a spike. This is called "anode break excitation."