I know that neutrinos are particles with a very small mass and no electric charge. They infrequently interact with matter and so they can give us information about the "old" universe. But how can they do it?
Note: I am not an expert on neutrinos, so if I have missed anything, someone please let me know.
For one thing, the knowledge of the mere existence of neutrinos is important for a full understanding of our universe. Also you may have heard that for a long time neutrinos were thought to be massless, but the observation of neutrino flavor oscillation requires them to have mass. This is a bit of a problem as the standard model of particle physics doesn't account for non-zero neutrino masses, which is an indication of some new physics, beyond the standard model.
As you mentioned, neutrinos are very weakly interacting with other matter, making it hard to detect them, which is why neutrino detectors are built on such large scales (see http://icecube.wisc.edu and http://www-sk.icrr.u-tokyo.ac.jp/kam/kamiokande.html for example). But this can be useful in terms of things we may want to find out about our universe. This very weakly interacting behavior enables physicist to see the neutrino spectrum from a star, supernova, gamma ray bursts, etc. virtually unchanged on the way from the object to Earth. Meaning that information about objects very far from Earth (and therefore far back in time) can be found where conventional techniques using light are less successful (for example if there is dust or some other object in between Earth and the object of interest - the light would interact whereas the neutrinos pass through pretty much unchanged).
In the "old" universe there is interaction of matter by the the 4 fundamental forces. Neutrinos occur in the weak nuclear interactions. By studying the neutrinos one can study these weak interactions. They might not interact much, but they sure do carry energy.
So by a application of the conservation of energy one could determine the loss to neutrinos and try to link these to the interactions that they might think that occur. For example the nuclear fusion in the sun that converts hydrogen into deterium:
gives an electron-neutrino. These neutrino's don't interact and the amount of energy of the neutrino is hence determined by the conservation of energy.
One could construct a model for the "old universe" and determine the energy of the neutrinos, as the missing energy. This could give some extra information I believe.