Getting max value from rows and joining to another table

What you wrote was missing A in the from clause so its not entirely clear where you went wrong but this should work

select 
       B.Id, 
       B.FileName,
       A.Name
FRom B
     INNER JOIN A
     ON A.id = B.id
    INNER JOIN ( 
          select A.Id, MAX(A.Rank)as ExpertRank 
          from A 
          group by A.Id
     ) as NewA 
    ON a.Id = NewA.ID 
       AND a.Rank = NewA.ExpertRank

See it working here

Alternatively you could use rownumber instead

WITH CTE AS 
(
   SELECT ID, 
          RANK,
          Name,
          ROW_NUMBER() OVER (PARTITION BY ID ORDER BY RANK DESC) rn
   FROM A
)
SELECT b.Id b.FileName,cte.Name
FROM
   b
   INNER JOIN cte 
   ON b.id = cte.id
      and cte.rn = 1

See it working here

Here's an answer with JOINs instead of MAX():

SELECT DISTINCT b.id, b.filename, a1.name
FROM a a1
JOIN b
  ON b.id = a1.id
LEFT JOIN a a2
  ON a2.id = a1.id
  AND a2.rank > a1.rank
WHERE a2.id IS NULL

If there are no duplicate ranks for the same id, then you don't need the DISTINCT.

personally, i like the below. just cleaner imo (though not sure about performace)

select * 
from a
inner join b on a.id=b.id and b.rank=(select max(rank) rnk from b where id=id)