How can i speed up this Indexed View?

What edition of SQL Server? I believe that only Enterprise and Developer Edition will use indexed views automatically, while the others support it using query hints.

SELECT a.PostId
FROM PostsCleanSubjectView a WITH (NOEXPAND)
WHERE a.CleanedSubject = 'Just-out-of-town' AND a.PostTypeId = 1

From Query Hints (Transact SQL) on MSDN:

The indexed view is not expanded only if the view is directly referenced in the SELECT part of the query and WITH (NOEXPAND) or WITH (NOEXPAND, INDEX( index_value [ ,...n ] ) ) is specified.

I see an @ sign in the query code in your execution plan. There's a string variable involved.

Sql Server has a NASTY behavior if the string variable's type does not match the type of the string column in the index. Sql Server will ... convert the whole column to that type, perform the fast lookup, and then throw away the converted index so it can do the whole thing again next query.

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Simon figured it out - but here's more helpful detail: http://msdn.microsoft.com/en-us/library/ms187373.aspx

If a query contains references to columns that are present both in an indexed view and base tables, and the query optimizer determines that using the indexed view provides the best method for executing the query, the query optimizer uses the index on the view. This function is called indexed view matching, and is supported only in the SQL Server Enterprise and Developer editions.

However, for the optimizer to consider indexed views for matching or use an indexed view that is referenced with the NOEXPAND hint, the following SET options must be set to ON:

So, what's happening here is that indexed view matching is not working. Make sure you're using Enterprise or Developer editions of Sql Server (pretty likely). Then check your SET options according to the article.