C. Palm's theory of spatial point processes relies heavily on measure theory in an abstract setting. A more gentle introduction is given in the lecture notes Conditioning in spatial point processes. Section 3, in particular, defines the Palm distribution on a finite set of points. This definition involves only elementary concepts.
I now notice that these lecture notes are the same source as mentioned in the OP (without a link, so I did not notice earlier). I do think this is the most accessible exposition of the Palm distribution, by considering the finite case first.