This terminology was common in the 1990s, before AI meant "machine learning" (aka adjusting parameters). Symbolic processing involves having an object in memory for things in the real world, and manipulating them. There's some vague relation to object oriented programming. The point is, it involves creating & destroying objects, memory allocation, that sort of thing. As opposed to purely numerical computing: for a neural net with a fixed architecture, there are a fixed number of operations (multiplying, adding) to get an answer. It's easy to do that in real time, since the computation is not only bounded, but of constant time. Symbolic manipulations can take variable amounts of time, and depend on the particulars of the inputs & problem being solved.

Lisp was championed by people in AI working on symbolic reasoning back in the 1990s.