You didn't actually run it - you just created a procedure.
Could've been like this:
Sample table:
SQL> CREATE TABLE AUDIT_20211031
  2  AS
  3     SELECT 1 act_code, 'Littlefoot' change_by, SYSDATE change_date FROM DUAL;
Table created.
Procedure (it is a good habit to display statement you'll run using dbms_output.put_line; once you make sure it is OK, remove that line):
SQL> CREATE OR REPLACE PROCEDURE GetLatestMonthAuditTable
  2  AS
  3     v_lastmonth  DATE := INTERVAL '-1' MONTH + SYSTIMESTAMP;
  4     v_year       VARCHAR2 (4) := EXTRACT (YEAR FROM v_lastmonth);
  5     v_month      VARCHAR2 (2) := EXTRACT (MONTH FROM v_lastmonth);
  6     v_day        VARCHAR2 (2) := EXTRACT (DAY FROM LAST_DAY (v_lastMonth));
  7     v_sql        VARCHAR2 (200);
  8  BEGIN
  9     v_sql :=
 10           'SELECT ACT_CODE, CHANGE_BY, CHANGE_DATE FROM AUDIT_'
 11        || v_year
 12        || v_month
 13        || v_day;
 14
 15     DBMS_OUTPUT.put_line (v_sql);
 16
 17     EXECUTE IMMEDIATE v_sql;
 18  END;
 19  /
Procedure created.
Testing:
SQL> SET SERVEROUTPUT ON
SQL>
SQL> BEGIN
  2     GetLatestMonthAuditTable;
  3  END;
  4  /
SELECT ACT_CODE, CHANGE_BY, CHANGE_DATE FROM AUDIT_20211031
PL/SQL procedure successfully completed.
SQL>
Now, your procedure doesn't do anything - it runs that select, but it isn't displayed anywhere on the screen.
If it were a function instead, you could return ref cursor and see something. For example:
SQL> CREATE OR REPLACE FUNCTION GetLatestMonthAuditTable
  2     RETURN SYS_REFCURSOR
  3  AS
  4     v_lastmonth  DATE := INTERVAL '-1' MONTH + SYSTIMESTAMP;
  5     v_year       VARCHAR2 (4) := EXTRACT (YEAR FROM v_lastmonth);
  6     v_month      VARCHAR2 (2) := EXTRACT (MONTH FROM v_lastmonth);
  7     v_day        VARCHAR2 (2) := EXTRACT (DAY FROM LAST_DAY (v_lastMonth));
  8     v_sql        VARCHAR2 (200);
  9     rc           SYS_REFCURSOR;
 10  BEGIN
 11     v_sql :=
 12           'SELECT ACT_CODE, CHANGE_BY, CHANGE_DATE FROM AUDIT_'
 13        || v_year
 14        || v_month
 15        || v_day;
 16
 17     OPEN rc FOR v_sql;
 18
 19     RETURN rc;
 20  END;
 21  /
Function created.
Testing the function:
SQL> SELECT GetLatestMonthAuditTable FROM DUAL;
GETLATESTMONTHAUDITT
--------------------
CURSOR STATEMENT : 1
CURSOR STATEMENT : 1
  ACT_CODE CHANGE_BY  CHANGE_DATE
---------- ---------- -------------------
         1 Littlefoot 16.11.2021 12:14:18
SQL>