One of the difficult part of reading the press articles is to understand what the scientific really saw in the experiment.

The original paper is https://arxiv.org/abs/1504.01527 The important graph is figure 4 in page 4. I made a colored version: http://imgur.com/Mya66KZ

The black circles are the experimental data. That's what they really saw. (Actually, these measures are very indirect and noisy, so this is the data after a lot of processing, but let's assume that this is standard incontrovertible processing.)

Each measurement has an experimental error, that is marked by the + cross in each circle. So imagine that each circle is a bigger fuzzy ellipse that is 7 times wider and 3 times taller than the circles.

If they assume that there is no new particle, in their simulation they would get the dashed curve, that is always decreasing. It's dashed in the original graph and painted in gray in the "improved" version.

The problem is that the experimental data in the ellipses are too far from the theoretical curve of the simulation without a new particle, so something strange is happening. (A new particle, experimental error, other ...)

They assume that the difference is caused by a new unknown particle, with some properties. In particular, they have to guess the mass of the particle.

If they guess that the particle is very heavy, for example 17.6MeV, they get a curve that looks like an "U", it's dot-dashed in the original and painted in blue in the "improved" graph. It coincides with the "no particle" curve when the angle is less than 130°, but then it starts increasing. The new curve doesn't look like the experimental ellipses at all, so they have to keep trying.

If they guess that the particle is very light, for example 15.6MeV, they get a curve that is also decreasing, it's dotted in the original graph and painted in yellow in the "improved" graph. It coincides with the "no particle" curve when the angle is less than 110°, but then it is bigger. But comparing it with the experimental data, it is too big for 110°-130° and too small for 130°-160°. The new curve also doesn't look like the experimental ellipses at all, so they have to keep trying.

If they guess that the mass of the particle is 16.6MeV, they get a curve with a bump, it's solid in the original graph and painted in red in the "improved" graph. It coincides with the "no particle" curve when the angle is less than 110°, it's slighter bigger at 120° and has an interesting maximum near 140°. The new curve pass through all but the last ellipse, so it looks like a good fit. So the conclusion is that they probably have a new unknown particle with a mass of 16.6MeV!!!

Actually, they (obviously) tried with more possible masses, not only this three values. And compare the simulation with the data using statistics, not the drawings. The details are in figure 5. With 16.6MeV they got the better agreement with the experimental data.

The particles have other properties besides mass. Some of them can be calculated with theoretically, for example the charge. Others, like spin, have to be tested using simulations, because using other values produces different looking curves.

Obviously, the complete paper has more details, but I hope this is a good starting point to start reading it.

EDIT: Fixed "GeV" -> "MeV"