You don't need to 'supply' neutrons. The nuclei of the Uranium 235 in the reactor are continually breaking down at random and spontaneously, at a low rate. This produces fast neutrons. If a fast neutron hits another U235 nucleus it will cause it to split and produce more neutrons. If these all hit other nuclei and all of these hit more nuclei, you have a chain reaction. If the Uranium has too little U235, the process won't take off because most of the neutrons will slow down and become harmless before they have hit U235 nuclei. You need sufficient Mass (the critical mass) of U235 in a sufficiently small space (i.e. high enough purity Uranium) for this to happen. With not enough, too many neutrons get lost around the edges.
The secret, in a reactor, is to arrange for the situation to be just above critical so that there are enough neutrons produced to keep the process going. This is done by having the fuel rods spaced in regular rows and to raise and lower 'control rods' which can absorb the neutrons or let them through to control the rate of the reaction and get it just as you want it.
The problem is that you can't stop this reaction very quickly. Even when you drop in the control rods there is still a lot going on in there and heat is still being produced. The reactors, in Japan, were shut down as soon as needed, I believe, but then other problems meant that there wasn't enough water available to cool everything down. This produced damage and even more problems etc. etc..