Auto logging / tracking records changes at MS SQL
There's a few different options available to do this, and also depend on which version of SQL Server you're running. Here they are in order of least complicated to implement and most standard, in my opinion:

Temporal Tables - These generate a log table for each table you want to track changes on, with who, when, and what the change was.

Change Data Capture - Similar to Temporal Tables, but on a more global level. System tables of the same database are used to log the changes of your tables within that database.

Database DML Triggers - You can create an AFTER TRIGGER that manually logs the changes into a separate table. This is more of a homebrew solution and a little more work than using either of the first two features.

Audit - This globally tracks multiple events, a subset of which would be DML changes such as UPDATES and INSERTS.