What has been measured at CERN some days ago?
Once you have an event (as @user1504 describes in his answer), i.e. a proton proton interaction, and have the four vectors of all the interaction products, i.e. jets, photons, leptons, you can generate invariant masses of the interaction products. The Higgs particle has been predicted theoretically and is the last piece in the jig saw puzzle that the Standard Model has put together in one picture. The invariant mass of two photons shows an enhancement consistent with the Higgs properties. Also the invariant masses of Z Z to a lesser extent. The combined statistics of all possible decay channels of the Higgs seen give a 5sigma certainty that the resonance is there and has the expected decay behavior within statistics of the individual decay channels.
What are the essential ingredients of the theory necessary to interpret said measurement? And so how do we deduce from the results that there is a new field/particle observed?
The theoretical part is a different question and should be posed independently: why and how the standard model theory accommodates all particle data.
That there is a new particle observed with 5 sigma certainty is what the data tells us, and it is new because we have never observed a 125GeV resonance before the present experiments.
To gauge whether it is the expected from the SM theory Higgs particle one needs good statistics in all the decay channels plus angular distributions that will establish the spin parity. Th theoretical Higgs has 0 spin and positive parity.
How to read the most relevant graphs in the presentation of the results?
The resonance is clear in fig 3. The sigma of this channel is its statistical significance for it alone. Similar for the ZZ channel, fig 4.
Fig 5 gives the probability,
The observed probability (local p-value) that the background-only hypothesis would yield the same or more events as are seen in the CMS data, as a function of the SM Higgs boson mass for the five channels considered. The solid black line shows the combined local p-value for all channels.