WHERE attribute having two relation lines in the same SQL
Use a subquery that has the checklist id that has both tags

SELECT
    B.id checklist_id,
    B.name checklist_name,
    D.name tag_name
FROM
(
    SELECT AA.checklistId,COUNT(1) tagcount
    FROM checklist_tag AA
    INNER JOIN tag BB ON AA.tagId = BB.id
    WHERE BB.name IN ('tag 1', 'tag 2')
    GROUP BY AA.checklistId HAVING COUNT(1)=2
) A
INNER JOIN checklist B     ON A.checklistId = B.id
INNER JOIN checklist_tag C ON A.checklistId = C.checklist_id
INNER JOIN tag D           ON C.tag_id = D.id;