The energy spectrum of the cosmic radiation (not CMB) is limited to both sides.
I know about the GZK-cutoff at high energies. Basically the interaction probability for photons of energies above 10^20 eV becomes so high, that all have interacted before they can reach us.
But why is there a limit at lower energies? Earth magnetic field and/or athmosphere, radiation belt? Perhaps someone can explain that to me.
I am not exactly sure which low energy cutoff you refer to; however, there is a low-energy cutoff for photons that I am aware of. Photons with energies on the order of $H_0\sim10^{-33}\text{eV}$ would be super-horizon modes. That is, their wavelengths would be on the order of the Hubble radius, $H_0^{-1}=14.6~Gly$. Larger than this would mean that the photon's peaks are essentially acausal as they would fall outside the current comoving horizon of every other peak. As such, we also could not measure such signals; because we could not see the full wave or the periodicity, it would look like preferred orientations of the EM fields on those scales.
But why is there a limit at lower energies?
Cosmic rays below ~10 GeV are affected by solar activity, specifically coronal mass ejections or CMEs. These are large-scale (i.e., they expand nearly as fast as they propagate), magnetically enhanced clouds of ionized gas called plasma that are ejected from the sun by processes including magnetic reconnection.
The number and strength of CMEs is strongly linked to the solar cycle. Thus, there is an anti-correlation between the cosmic ray fluxes below ~10 GeV and the sunspot number. This is called the Forbush decrease. This is discussed in more detail in the answer to the following question: How does the 11-year solar cycle alter the cosmic ray flux?.
Earth magnetic field and/or athmosphere, radiation belt?
No, the low energy decrease is seen by spacecraft (e.g., ACE) outside the Earth's magnetosphere, i.e., far away from both the atmosphere and radiation belts.
In a talk a lower limit at about 1GeV was mentiond as a side note.
The Forbush decrease is likely to what the speaker was referring.
Your question covers a number of different topics.
The GZK process refers to hadrons not photons. Also, it isn't exactly some kind of hard cut-off. The mean free path is on the order of several Mpc (which is comparable to the distance between neighboring galaxies) and even then only 20-50% of the energy is lost at each interaction. So particles with $E>E_{GZK}$ will still traverse quite a distance.
On the lower end, protons can go as slow as they like (I feel like I'm going pretty slow) which brings up the question of relativity. The speed of the proton only depends on some reference frame which, for the GZK effect, is the CMB.
As for other low energy particles there can be some low energy effects depending on the particular conditions, but nothing as general as the GZK against the CMB. See here (wikipedia).
