Well, don't just yank on the door handle; that's a death sentence.
See? I told you so. See that odd contraption on the right side of the doors? Look closely to see that it's some sort of scale. That will eventually help you open the doors.
After you follow up on the first weird note --
-- which leads you back here, where your goal is to put all the seeds on the scale in such a way that the scale balances. Read to the end of this hint for the way to place the seeds.
The numbers on the little trays tell you the weight of each seed placed there. The scales will balance when you have the same number on each side of the scale.
For example, if you put one seed in the 5 tray and one seed in the 4 tray, that's 5 + 4, or 9. If you have two seeds in the 5 tray (2 X 5 = 10) and three seeds in the 4 tray (3 X 4 = 12), you'd have a total of 22 (10 + 12 = 22). See?
So this is just basic arithmetic after all --
Follow the new tunnel down to find Jess's backpack, which is full of neat stuff --
-- and open the door beyond that to find the cellar, which is likewise full of interesting things.