SQL Server Agent logging actions and changes done to jobs
For sake of simplicity, I will assume that you want to track sysjobs, sysjobsteps and sysjobschedules. There may be other tables you want to monitor.

Option 1: SQL Audit (requires Enterprise Edition)

USE [master]
GO

-- Audit
CREATE SERVER AUDIT [jobs]
TO FILE 
(   FILEPATH = N'PathToSomeFolder'
    ,MAXSIZE = 0 MB
    ,MAX_ROLLOVER_FILES = 2147483647
    ,RESERVE_DISK_SPACE = OFF
)
WITH
(   QUEUE_DELAY = 1000
    ,ON_FAILURE = CONTINUE
    ,AUDIT_GUID = 'e807469a-6c9d-43f1-af46-cf7b89ba898d'
)
ALTER SERVER AUDIT [jobs] WITH (STATE = ON)
GO

USE [msdb]
GO

CREATE DATABASE AUDIT SPECIFICATION [job_changes]
FOR SERVER AUDIT [jobs]
ADD (UPDATE ON OBJECT::[dbo].[sysjobs] BY [public]),
ADD (UPDATE ON OBJECT::[dbo].[sysjobsteps] BY [public]),
ADD (UPDATE ON OBJECT::[dbo].[sysjobschedules] BY [public])
WITH (STATE = ON)
GO


The data captured by the audit appears like this:


Option 2: Extended Events session

-- Step 1: extract object_id for the following tables
SELECT object_id 
from sys.tables
WHERE name IN ('sysjobs','sysjobsteps','sysjobschedules');

-- Step 2: use those object_ids in the following session:
CREATE EVENT SESSION [capture_job_changes] ON SERVER
ADD EVENT sqlserver.lock_acquired (
    SET collect_database_name = (0)
        ,collect_resource_description = (1)
    ACTION(sqlserver.client_app_name, sqlserver.is_system, sqlserver.server_principal_name)
    WHERE (
        [package0].[equal_boolean]([sqlserver].[is_system], (0)) -- user SPID
        AND [package0].[equal_uint64]([resource_type], (5)) -- OBJECT
        AND [package0].[equal_uint64]([database_id], (4))  -- msdb
        AND (
               [object_id] = 1125579048 -- sysjobs
            OR [object_id] = 1269579561 -- sysjobsteps
            OR [object_id] = 1477580302 -- sysjobschedules
        )
        AND (
               [mode] = (8) -- IX
            OR [mode] = (5) -- X
        )
    )
)
WITH (
     MAX_MEMORY = 20480 KB
    ,EVENT_RETENTION_MODE = ALLOW_MULTIPLE_EVENT_LOSS
    ,MAX_DISPATCH_LATENCY = 30 SECONDS
    ,MAX_EVENT_SIZE = 0 KB
    ,MEMORY_PARTITION_MODE = NONE
    ,TRACK_CAUSALITY = OFF
    ,STARTUP_STATE = OFF
);
GO

-- Step 3: add a convenient target to the session (file target?)


The data in the XE session appears like this:


Regarding this second option, I wrote a blog post on a similar subject (tracking object usage) where I describe the details of the technique. Basically, you can consider IX/X locks as updates to the underlying tables.

This session captures the bare minimum, but you can add more fields/actions to it to capture the sql text or the computer name or whatever makes sense for you.