Since porphyrin covered the general case of a correlation function, I'll try to give a little more detail on what they were specifically modelling in this paper.
The correlations functions they define in the paper are correlations of indicator variables (a value of 1 when some criteria is true, and zero otherwise). The continuous correlation function gives the probability that a given hydrogen bond at time 0 remains all the way until a time t. The intermittent correlation function gives the probability that a given hydrogen bond at time 0 is also present at time t, even if the bond breaks and reforms in the time in between.
That these functions decay over time is a consequence of how these bonds fluctuate over time. If t=0, the correlation function is at a maximum, in this case because we know the hydrogen bond is still there. Shortly after, the function is still fairly because the bond hasn't had a lot of time to separate. At very long times, it is unlikely that particular hydrogen bond has remained in place or even reformed.