That feature is called a common table expression
http://msdn.microsoft.com/en-us/library/ms190766.aspx
You won't be able to do the exact thing in mySQL, the easiest thing would to probably make a view that mirrors that CTE and just select from the view. You can do it with subqueries, but that will perform really poorly. If you run into any CTEs that do recursion, I don't know how you'd be able to recreate that without using stored procedures.
EDIT:
As I said in my comment, that example you posted has no need for a CTE, so you must have simplified it for the question since it can be just written as
SELECT article.*, userinfo.*, category.* FROM question
INNER JOIN userinfo ON userinfo.user_userid=article.article_ownerid
INNER JOIN category ON article.article_categoryid=category.catid
WHERE article.article_isdeleted = 0
ORDER BY article_date DESC Limit 1, 3