Here's a simplified calculation to show the connection between the properties of nuclei, as shown in the Nuclear Table, and the occurence of a chain reaction in fission. There is a chain reaction because, when a heavy nucleus fissions into two medium-sized nuclei, several extra free neutrons are produced.
The Nuclear Table shows the following property, having to do with the number of neutrons (N) in the nucleus relative to the number of protons (Z):
Take the following as a simplified version of this relation:
Now suppose the heavy nucleus has Z = 100. (These are ficticious nuclei, not the ones in the Nuclear Table.) Therefore it has N = 150, and A (its mass number) is 250. Suppose it fissions into two medium nuclei with atomic numbers 60 and 40. The nucleus with Z = 60 must have N = 78 (30% more than 60). The nucleus with Z = 40 must have N = 52 (30% more than 40).
The heavy nucleus has 150 neutrons, but the two medium nuclei have a total of 130 neutrons (78 + 52). Therefore, there must be 20 extra neutrons somehow accounted for following the fission. They are the "free" neutrons, the ones that go on to cause fissions in other heavy nuclei.
To be more realistic, bear in mind that the 30% figure for medium nuclei represents a property of the common isotopes, the ones that are found in nature. The nuclei produced as the result of fission are not necessarily the same isotopes as those found in nature; they are likely to contain more neutrons than given by the 30% figure. Nevertheless, there will probably be some free neutrons produced, even if it's not as many as the 20 in the calculation above.