How are the chemical reactions of the table of standard reduction potentials experimentally determined?
The standard reduction potentials were mostly discovered by carefully measuring the reduction and oxidation of solid metals (or dissolved ions) in electrochemical cells, and monitoring the electrical current through the wire connecting the two electrodes. (The image below should have an ammeter instead of of a voltmeter, but you get the idea)

Take, for example, an electrochemical cell of zinc and copper.



At the anode, the solid zinc will dissolve to become Zn2+. At the cathode, the Cu2+ ions will be reduced, and become solid copper. During the redox reaction, the scientist measures the current across the two electrodes.

After some time, the scientist could pull the two electrodes out of solution, dry them off and mass them. They could then calculate the moles of zinc that dissolved from the anode, and the moles of copper added to the cathode. In addition, by measuring the current over the period of the experiment, they would also be able to calculate the number of electrons that passed from the zinc anode to the copper cathode. 

In this instance, they would calculate that for every mole of zinc that dissolved, 2 moles of electrons were produced. Concurrently, 2 electrons were consumed for each mole of copper reduced! These measurements, over time, made it possible to determine the oxidation states of many metals and soluble ions.

(image from moodle2.rockyview.ab.ca/)