This redox reaction occurs in acidic solution, meaning that we need to add H+ and H2O to help balance it out (for comparison, balancing in basic solution would involve OH- and H2O instead).
Notice that the right side of the half reaction already has oxygen added to it, so we need to add the water to the other side. This way, the left side of the reaction would get enough oxygen to match the right side. Add enough water to the left to match the number of oxygens on the right.
2H2O + Re — ReO2
We now have two oxygens on each side, but the left has four more hydrogens than the right. Fix this by adding 4H+ to the right side of the equation.
2H2O + Re — ReO2 + 4H+
The number of each element matches the number of each on the other side; the only thing left is to balance the charges. The left side of the equation has a charge of 0, but the right has a charge of +4. We can fix this by adding four electrons to the right to cancel out the +4 charge.
