Prior to 1932, the accepted model of the atom was that it contained only positively charged protons and negatively charged electrons.. Physicist Sir James Chadwick had noticed a discrepancy between the atomic numbers of atoms and the atomic weight. For example, the element Helium should have an atomic mass of 2 with the model of the atom at the time, it instead has a mass of 4. The reason this was discovered is that the element Hydrogen has one electron and one proton, with 1 being both the atomic number and atomic mass. Helium should have had twice the mass of Hydrogen, but instead had 4 times the mass.
The postulation that Chadwick came to is that there were two extra particles in a Helium atom. Demonstrating this would prove to be quite difficult because particle detection in that day was reliant on their electrical charges. Finding and demonstrating the existence of an electrically neutral particle would take an innovative technique.
Back in 1930, experiments were done on the nuclei of light elements where there was a neutral emission while these nuclei were bombarded with certain types of energies. This was seen by Chadwick as more evidence of a neutral particle in the nucleus. Because there were numerous reports of new particles that turned out to be incorrect, many of Chadwick's peers did not adhere to his neutral particle theory. It was mainly assumed that the neutral radiation from the nucleus were actually Gamma rays, which are neutral, high energy photons.
Since Gamma rays have no mass, Chadwick set out to prove that this neutral radiation had mass. Bombarding base elements with other high energy particles would cause them to break apart. When they break apart, their constituent particles would then collide with the other particles. By measuring the velocity of the struck particles, Chadwick was able to determine that the mass of the radiation was more than that of a photon, and was actually extremely close to that of a proton. The newly discovered neutron had a mass within 1% of the proton.
This experiment can be likened to pool balls on a pool table. If you were to strike an 8 ball with a marble, the 8 ball will not travel far due to the large difference in mass. Now strike the 8 ball with the cue ball, which has a very similar mass, and you can easily send it traveling far across the table. In this case, the 8 ball would likely be a proton from an element, the marble would represent a gamma ray or photon, and the cue ball would be the neutron. If the marble and the cue ball were invisible, you would prove that the cue ball existed by how far it moved the 8 ball when it struck. That is how Sir James Chadwick discovered, and proved the existence of, the neutron.