Whilst they are sterile and thus non interacting with anything, sterile neutrinos nevertheless mix a little with all other neutrinos. Since all other neutrinos are sensitive to the weak force, sterile neutrinos have hence a reduced sensitivity to the weak force through this mixing. This sensitivity is however super-suppressed because of the small size of the mixing.

Some sterile neutrinos could have been produced in the early universe, but also later on during the course of the universe history, through oscillations of non-sterile neutrinos.

Those oscillations means that when a neutrino propagates, there is a given probability that if it is of type A (A can be any type of neutrino), it will be detected as a neutrino of class B (again, B can be any type of neutrino). Emphasising with an explicit example: an electronic neutrino produced in a star could reach Earth as a muon, tau or even a sterile neutrino (assuming sterile neutrinos exist). Is that clearer?

Concerning the image, it shows the relations between the mixing pattern of the sterile neutrino with the electron and muon neutrinos (x-axis) and the mass difference between the sterile neutrino and the others (the y-axis). The coloured areas are the regions favoured by the anomalies, the red area being the most favoured one. The vertical lines (OPERA and ICARUS) are constraints: everything on their right is excluded. The other lines represent constraints from the non observation of anything by other experiments: everything on top is excluded. I hope this clarifies the whole story. Otherwise, feel free to come back to me!