Your expression in a WHERE clause works against a single row of the joined result set. That’s why
WHERE category_id = 201 AND category_id = 202 doesn’t work — because it can’t be two values on a single row.
So you need some way to join two rows from the table into one row of the result set. You can do this with a self-join:
FROM item_category AS c1
INNER JOIN item_category AS c2 ON c1.item_id = c2.item_id
WHERE c1.category_id = 201 AND c2.category_id = 202
This technique is hard to scale up when you want to search for three, four, five or more values, because it requires N-1 joins to match N values.
So another method is to use GROUP BY:
SELECT c.item_id, COUNT(*) AS cat_count
FROM item_category AS c
WHERE c.category_id IN (201,202)
GROUP BY c.item_id
HAVING cat_count = 2
Both techniques are okay, and work better in different circumstances.