Why does each node in the wavelet scattering transform split into multiple paths?
We take a wavelet transform over other wavelet transforms, which involves multiple wavelets. E.g.:

First-order uses 100 wavelets to tile the frequency domain and produce 100 coefficients
Second-order uses 16 wavelets to tile the frequency domain and produces 8 coefficients per first-order coefficient, yielding a total of 800
Third-order uses 9 wavelets ... etc

(Note second-order wouldn't actually produce 800 in implementation since xi2 &gt; xi1 wavelets are discarded, where xi == center frequency).
The idea is, first-order coefficients capture temporal modulations (AM-FM). Second-order capture the modulations of those modulations, extracting higher order structure, and the process repeats. Progressively we require less wavelets as there's less fine-grained structure to capture (also energy and non-redundancy reasons), but it's still more than one wavelet (i.e. path), as just one wavelet cannot cover entire frequency domain without losing information (unless it's very wide, but that's a bad wavelet).
See helpful lecture, and Python implementation. I also might write a detailed post on this topic in coming weeks.