What is a cursor? Why not just SELECT into variables?

Single-row SELECT INTO makes sense. For many rows people say DECLARE CURSOR / OPEN / FETCH / CLOSE.

Is a cursor just a pointer walking the result set?

Yeah. SELECT INTO expects one row (or you handle -811). Cursor is how batch COBOL walks many rows: OPEN, FETCH until SQLCODE 100, CLOSE.

Remember to check SQLCODE on each FETCH.