In your trigger, you have two pseudo-tables available, Inserted and Deleted, which contain those values.
In the case of an UPDATE, the Deleted table will contain the old values, while the Inserted table contains the new values.
So if you want to log the ID, OldValue, NewValue in your trigger, you’d need to write something like:
CREATE TRIGGER trgEmployeeUpdate
ON dbo.Employees AFTER UPDATE
AS
INSERT INTO dbo.LogTable(ID, OldValue, NewValue)
SELECT i.ID, d.Name, i.Name
FROM Inserted i
INNER JOIN Deleted d ON i.ID = d.ID
Basically, you join the Inserted and Deleted pseudo-tables, grab the ID (which is the same, I presume, in both cases), the old value from the Deleted table, the new value from the Inserted table, and you store everything in the LogTable