The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.

Optimal indexes for this query:

CREATE INDEX favourites_idx_articlesuserid_articleid ON "favourites" ("articlesUserID","articleID");
CREATE INDEX posts_idx_userid_postdate ON "posts" ("userID","postdate" desc);

The optimized query:

SELECT
        posts.id,
        posts.title,
        posts.author,
        posts.postdate,
        posts.postcontent,
        posts.userID 
    FROM
        posts 
    WHERE
        posts.userID = 12 
        AND EXISTS (
            SELECT
                1 
            FROM
                favourites 
            WHERE
                (
                    favourites.articlesUserID = 12
                ) 
                AND (
                    posts.id = favourites.articleID
                )
        ) 
    ORDER BY
        posts.postdate DESC