I believe you can do it something like this.

with ttt as (

select 123 Id, DATEADD(d,-10,getdate()) fromDD, DATEADD(d,-3,getdate()) toDD,

datediff(d, DATEADD(d,-10,getdate()), DATEADD(d,-3,getdate())) period ,1000 prodID union

select 202 Id, DATEADD(d,-25,getdate()) fromDD, DATEADD(d,-17,getdate()) toDD,

datediff(d, DATEADD(d,-25,getdate()), DATEADD(d,-17,getdate())) period ,2000 prodID union

select 303 Id, DATEADD(d,-55,getdate()) fromDD, DATEADD(d,-10,getdate()) toDD,

datediff(d, DATEADD(d,-55,getdate()), DATEADD(d,-10,getdate())) period ,3000 prodID

),

Tally (n) AS (

SELECT number

FROM [master].dbo.spt_values Tally

WHERE [Type] = 'P' AND Number BETWEEN 1 AND 100)

SELECT id, fromDD=MIN(d.DD), toDD=MAX(d.DD), period=COUNT(*), prodID

FROM ttt a

CROSS APPLY (SELECT n FROM Tally WHERE n BETWEEN 1 AND period+1) c

CROSS APPLY (SELECT [month]=DATEPART(month, a.fromDD+n-1),DD=CAST(a.fromDD+n-1 AS DATE)) d

GROUP BY id, prodID, [month]

Note that if you have a period > 100 days, then you'll need a bigger Tally table (examples abound). You can also do this with a Calendar table but that might be a little harder to explain.

Not sure if you really need that last CAST to DATE but you did show your expected results as a DATE and not a DATETIME.