This is the same answer I included in my solution for the "Completionist!" achievement.
The "Explorer" achievement can be a little tricky, it might be bugged. On my third playthrough I finally got this achievement although I've been through every hidden wall before (more than once!). So, if you face this problem like I did, just keep playing the game and enter every hidden area over and over again. This might be again related to what NO1KEV said, but I always completed the stage and got the "saved" sign on the loading screen.
It is important to point out that ALL hidden areas count, not only the ones containing upgrades. A few hidden areas will only give you health and armor, but these do count as well for the "Explorer" achievement so it is necessary to visit those too.