Group by without temporary table and filesort
Is ID_Links_Words a many-to-many table for connecting Words and Links?  Then this is the optimal schema:

CREATE TABLE ID_Links_Words(
    word ... NOT NULL,
    link ... NOT NULL,
    count ... NOT NULL,
    PRIMARY KEY(word, link), -- for going one direction
    INDEX(link, word)        -- and the other
) ENGINE = InnoDB;           -- Important because of the clustering of the PK.


The two indexes can be swapped, depending on the various queries you do; this arrangement -- with word first in the PRIMARY KEY -- is best for this SELECT.

Remove LEFT unless there is a compelling reason for it.

What percentage of the table has those 3 'words'?  If it is a non-trivial percentage, you are stuck with a table scan.

Check the settings of innodb_buffer_pool_size and key_buffer_size.

What version of MySQL are you using?  There were improvements in the handling of IN(...) in 5.6, perhaps before.  If you are in an older version, recommend upgrading.

If the my other hints don't do enough, try this:

(   SELECT  SQL_NO_CACHE L.id, L.link, sum(W.count) AS wordCount
        FROM  Links AS L
        LEFT JOIN  ID_Links_Words AS W ON L.id = W.link
        WHERE  W.word = '1025'
        GROUP BY  W.link 
)
UNION  ALL (
    SELECT  SQL_NO_CACHE L.id, L.link, sum(W.count) AS wordCount
        FROM  Links AS L
        LEFT JOIN  ID_Links_Words AS W ON L.id = W.link
        WHERE  W.word = '1056'
        GROUP BY  W.link 
           )
UNION  ALL (
    SELECT  SQL_NO_CACHE L.id, L.link, sum(W.count) AS wordCount
        FROM  Links AS L
        LEFT JOIN  ID_Links_Words AS W ON L.id = W.link
        WHERE  W.word = '2353'
        GROUP BY  W.link
           );


And, assuming that the LEFT is not necessary, test this

SELECT  W.link as id, 
        ( SELECT  link
            FROM  Links
            WHERE  id = W.link 
        ) AS link,
        wordCount
    FROM
      ( SELECT  SUM(count) AS wordCount
            FROM  ID_Links_Words
            WHERE  word IN ('1025','1056','2353')
            GROUP BY  link 
      ) AS W ;

This, if I wrote it right, has the following potential benefits:


The GROUP BY is done without being bothered by the other table.
The other table is done after shrinking the number of rows by the GROUP BY.


Again, the IN could be turned into a UNION.

Edit

There is a world of difference between INDEX(word, link) and INDEX(link, word).  INDEXes are ordered.  If you have a word and need to find the links associated with it, the former is much better.  If you have a link and want to find the words the latter is much better.  Imagine looking in a list of names ordered by (last_name, first_name), and looking for all the last names, given a first name.

Index cookbook