Its meaning should be something like "degree distribution seen from a random edge". Indeed, let us first think of an unoriented edge as of two oriented (in opposite directions) ones. Assume that there are $N$ nodes, where $N$ is large. Then, the total number of (oriented) edges should be roughly $N\sum_d d f(d)$. So, if you choose one edge at random and then ask "what's the probability that the degree of a node where this edge begins is $d$?", it is clear that this probability should be proportional to $df(d)$ (since there are $Ndf(d)$ such edges out of $N\sum_{d'} d' f(d)$).

Its meaning should be something like "degree distribution seen from a random edge". Indeed, let us first think of an unoriented edge as of two oriented (in opposite directions) ones. Assume that there are $N$ nodes, where $N$ is large. Then, the total number of (oriented) edges should be roughly $N\sum_d d f(d)$. So, if you choose one edge at random and then ask "what's the probability that the degree of a node where this edge begins is $d$?", it is clear that this probability should be proportional to $df(d)$ (since there are $Ndf(d)$ such edges out of $N\sum_{d'} d' f(d')$).