What is the purpose of learning propositional logic
Briefly, one can't learn AI without it.
Some of the main content areas of AI are based on logic: knowledge representation, planning, natural language semantics and reasoning. Have a look at the leading AI textbook to see this in detail. A lot of this uses first-order logic, but first-order logic and many of the other major forms of logic are built on propositional logic.
AI has either used or created several other kinds of logic: non-monotonic logic, temporal logic, fuzzy logic, intuitionistic logic, and modal logic. Check out this 5-volume set about logic in AI.
Back to propositional logic: in the 90's it was realized that SAT solvers (programs which could find satisfying assignments to most propositional logic formulas) could solve formerly intractable problems in AI planning. This has been an active research field ever since.
Propositional logic also has a number of very desirable properties: it is consistent, complete, sound, and decidable. Not all forms of logic have all these properties. For example, decidability breaks down in first order logic.
Finally, it's worth knowing that a lot of other stuff in computer science is based on propositional logic. Consider one of the fundamental theorems of complexity theory: the Cook-Levin theorem, which provides a theoretical basis of complexity theory by showing that bounded computations in a Turing machine can be modeled with propositional logic.