How to use a CTE statement in a table-valued function in SQL Server

Syntax for the CTE in table valued function would be:

CREATE FUNCTION GetDistributionTable 
(
    @IntID int,
    @TestID int,
    @DateFrom datetime,
    @DateTo datetime
)
RETURNS TABLE
AS
RETURN  
(
    WITH cte AS
    (
        SELECT ROUND(Result - AVG(Result) OVER(), 1) Result
        FROM   RawResults 
        WHERE  IntID = @IntID 
        AND    DBTestID = @TestID 
        AND    Time >= @DateFrom 
        AND Time <= @DateTo    
    )

    SELECT  COUNT(*) AS [Count],
            Result
    FROM    cte
    GROUP  BY 
            Result
)
GO

If possible, you can also omit the CTE (WITH statement), and instead create an inline table valued function that uses subquery:

CREATE FUNCTION GetDistributionTable 
(
    @IntID int,
    @TestID int,
    @DateFrom datetime,
    @DateTo datetime
)
RETURNS TABLE
AS
RETURN  
(
    SELECT  COUNT(*) AS [Count],
            Result
    FROM    (
                 SELECT ROUND(Result - AVG(Result) OVER(), 1) Result
                 FROM   RawResults 
                 WHERE  IntID = @IntID 
                 AND    DBTestID = @TestID 
                 AND    Time >= @DateFrom 
                 AND Time <= @DateTo    
    ) t
    GROUP  BY 
            Result
)
GO

Your example seems to be using a multi-statement TVF (insert and select), when you have a choice try using the inline TVF because the multi-statement TVF can prevent query optimizer in choosing a better execution plan (performance difference explained here)

LIKE THIS..

CREATE FUNCTION GetDistributionTable 
(
    @IntID int,
    @TestID int,
    @DateFrom datetime,
    @DateTo datetime
)
RETURNS 
@Table_Var TABLE 
(
    [Count] int, 
    Result float
)
AS
BEGIN
  WITH T 
    AS (    
        select Ticket_Id,COUNT(1) Result from 
        Customer_Survey
        group by MemberID,SiteId,Ticket_Id
   )
  INSERT INTO @Table_Var ([Count], Result)
  SELECT COUNT(*) AS [Count],
       Result
  FROM   T
  GROUP  BY Result
  RETURN 
END
GO