There are many ways to achieve what you want. Considering that your table has the following structure ..
CREATE TABLE blabla (user_id INT,activity_type VARCHAR,interested_rooms VARCHAR);
INSERT INTO blabla VALUES (1,'views','1,2'),
                          (1,'application','1'),
                          (1,'search','3,2'); 
.. you could try a CTE ..
WITH j AS (
  SELECT DISTINCT user_id, unnest(string_to_array(interested_rooms,',')) AS rooms
  FROM blabla ORDER BY rooms
)
SELECT user_id, array_agg(rooms) AS interested_rooms
FROM j GROUP BY user_id
.. that would return the result you want
 user_id | interested_rooms 
---------+------------------
       1 | {1,2,3}
Note that the usage of the functions unnest and string_to_array will depend on the data type of the column interested_rooms which wasn't provided in your question. But it should give you an orientation of what to do. Good luck!