The explanation consists of two short and simple stages:
(i) 3 cats + 4 cats = 7 cats.
The 7 came from (3+4), so we can write:
3 cats + 4 cats = (3 + 4) cats
We don’t need the whole cat, ‘C’ will do:
3 C + 4 C = (3+4) C (3 C is the same as 3×C)
This works just as well with any pair of numbers, say, 'n' and 'p' (instead of the 3 & 4), so:
n × C + p × C = (n+p) × C
Reversing the order, and using different letters:
C × (a+b) = C × a + C × b
The ‘×’ are sometimes omitted or replaced by a dot, i.e.
C(a+b) = C⋅a + C⋅b