Show me the products that are used on every Project in Athens
is equivalent to
Show me the products where there is no project in Athens that does not use that product.
This last sentence is exactly what the query above says.
SELECT p.pname -- Show me the products
FROM product p
WHERE NOT EXISTS -- where there is **no** project
(SELECT *
FROM project j
WHERE j.city = 'athens' -- in Athens
AND NOT EXISTS -- that does not use
(SELECT * FROM spj
WHERE spj.pid = p.pid -- that product
AND spj.jid = j.jid))