How can a one-counter machine accept $a^n b^n c^n$?
Be careful where the hyphens are! One counter-machine can recognize $a^n b^n c^n$, but a one-counter machine cannot.

From the answer you linked to:


  Languages recognized by one[-]counter automata form a proper subset of the context free languages.


(There's a proof for this, but it's long and boring, so I'm going to leave it out; you can find it online if you're interested.)

Since $a^n b^n c^n$ is not context-free, it cannot be recognized by a one-counter automaton.

It can, however, be recognized by a two-counter machine (a machine with two counters). The basic structure goes something like this:


When you see an $a$, increment $X$, increment $Y$
When you see a $b$, decrement $X$
When you see a $c$, decrement $Y$
Accept iff the letters are in the right order (a trivial state machine), AND $X = 0$, AND $Y = 0$


This is "one counter-machine" (in the sense that it's a single machine using counters). But you need at least two counters to do it.