After Update Trigger When Multiple Rows To Update
You are right, there is a virtual table called "inserted" which is what you would need to join on. This table contains all the rows which have been inserted in the original table insert statement. So something like:

CREATE TRIGGER [dbo].[UpdateFields]
ON  [dbo].[TFData]
AFTER INSERT
AS 
BEGIN
SET NOCOUNT ON;
UPDATE a
Set 
a.DateOfInjury = b.InjurDate
,a.GameNum = c.GameNum
FROM [dbo].[InjuryReport] a
INNER JOIN [dbo].[GameStats] b
ON a.playnum = b.playnum
INNER JOIN [dbo].[GameSched] c
On b.gamenum = c.gamenum
INNER JOIN inserted i
ON b.linkfield = i.linkfield