If the current is kept to zero then how do electrons move from one electrode to the other electrode? How would the reduction happen if electrons don't move from one electrode to the other electrode?
Zero in science is not taken as rigorously as in mathematics. It is seldom exactly zero, but rather any value smaller than some significant or noticeable threshold.
Scientific zero is in data processing context any value with its difference to mathematical zero being statistically not significant.
In the question context, the zero current means a current small enough not to noticeably disturbing electrode equilibrium.
It is comparable to a voltmeter measuring voltage on particular points of electronic circuits, that has ideally infinite resistance and draws zero current from the circuit. In reality, its resistance is finite and it does draws some current, that is considered negligible, taking as zero for circuit servicing purposes.
No current needs to flow from an open circuit half cell. Well, no external current needs to flow. An equilibrium is established between the solution and the electrode. This equilibrium involves many transitions, back and forth, but is in equilibrium. There is a net current of zero.
Now, how do you measure a voltage without getting a current? You back it up with an equal and opposite voltage so as to establish a current of zero. Now, I admit that zero is pretty difficult to hit, exactly, but you can come so close that whether the current in or out of the half cell is + or - an electron or two or three, it doesn't change the measured voltage significantly.
A Wheatstone Bridge is the way to get zero current; normally, the galvanometer is very sensitive and resistances to be measured are adjusted for zero current. https://en.wikipedia.org/wiki/Wheatstone_bridge
To measure the voltage of a half cell without drawing current, the half cell is placed in series with the galvanometer and the resistances are adjusted for zero current. Then half cell is disconnected and its voltage is read off the actually unbalanced bridge, which can provide enough current for a non-infinite input resistance voltmeter.