If you are new to SQL I can give you one piece of advice - a lot of the advice you get will be bad advice!
In this case, it is doubtful that you will get advantage from "using CTEs" - that is something that gets thrown around as if it were a magical fix for bad query performance.
In general, your query is fine.
You don't need to use expressions like
LEFT JOIN(SELECT Real_ID, Material FROM View.Real_Item_ID) as b in this query. Just tell SQL the tables you want to join and and let it build the query. There is no reason for the "premature selection" as SQL will be able to resolve selecting the needed columns after the joins are all resolved.
SQL is about DATA. You have to provide context for your queries in order to make good judgments. In this case, we don't know what the primary keys are in these tables. Is
Real_ID supposed to always match either
Real_ID ever null? Provide some sample data that covers all the possible cases for how these three IDs work together and explain the reason for the query. If all this information is missing or unknown, you have to write more conservatively (like assuming nulls are possible, duplicates are possible, missing data is possible ... when perhaps these things are not actually possible at all).
Everything must be tested. You can verify if a CTE works better or not by testing the result. There are three tables here. I don't see how a CTE can magically do this work with one join (because you need two joins to join three tables). Personally I don't see any obvious use of CTE's here that makes sense (well, this is three points, I guess).
Here's how I would write your query:
b.Real_ID as Real_ID_From_Material_Num,
c.Real_ID as Real_ID_From_Alt_Material_Num
#MaterialTable as a
left join #Real_Item_ID as b
on a.Material_Num = b.Material
left join #Real_Item_ID as c
on a.Alt_Material_Num = c.Material;