Start by sealing any cracks with weatherstripping. Then I'd suggest investing in some heavy curtains. You can also try hanging acoustical panels on the wall (either build them yourself, or find somewhere that supplies home theater systems).
There's two principles to soundproofing that actually work: Isolation and Absorption. The trick is figuring out the right way to implement them. Be careful of anything that doesn't follow these two principles.
The Absorption principle is how acoustic ceiling tiles and carpets work: They absorb the sounds as they pass through the material, or they keep the sound from making it into the intermediary material (the floor and frame of your house) in the first place.
The Isolation principle focuses on uncoupling the materials so that sound can't be transmitted through them and amplified. It's equivalent to rubber feet on appliances or rubberized engine mounts.
Note that drywall is a poor absorber. It's a reflector and can also be an amplifier. Speakers work by vibrating a large flat surface; in the same way that your kids vibrate the floor above which vibrates the frame of the house and the large flat expanse of drywall on the ceiling below... which amplifies the sound.
If you want to focus on absorption, a few ideas would include putting carpet down upstairs (or thicker padding), possibly attaching tapestries or acoustic tiling to the ceiling downstairs, and hanging draperies downstairs so that sound doesn't travel through the entire house. If you have wood floors, I'd start by putting down area rugs and runners on the stairs and hallways... that will be the most bang for your buck, and helps protect the floors from people walking above. This is also the least intrusive way of soundproofing your home.
Isolation involves a structural refit. You would need to tear the drywall off of the ceiling on the floor below, and put it back up with an uncoupling system that generally includes acoustic caulking and rubber washers to keep the drywall from communicating the sound from the wood framing it is attached to. This is obviously expensive... but when done right will provide a good solution. If you go all out, you'll do the floor above, too, but that's a huge mess that involves tearing up the subfloor, and it's probably not worth it.
Solutions that may provide a bit of a temporary fix but don't generally fix the problem include filling the cavities between floors with insulation. A thicker carpet would be a better fix, because with insulation you still have the frame of the house transmitting the sound to the drywall -- putting insulation in the cavity is equivalent to putting a pillow in a bass drum. It's still loud, just a little muted. Insulation is however valid if you were to have uncoupled the ceiling from the joists first.
Best Answer
.I assume that you will need to be able to use that passage and the walls as you said are soundproof enough. Then you need something like a door or a hanging curtain. To stop the sound you would need some mass and you can also add some absorption into your room. I suggest you start with the following: 1 - place the refrigerator on a rubber mat - that will decouple it from the floor and cut sound transmission this way; 2 - hang a heavy curtain. you can use mass loaded vinyl http://www.vocalboothtogo.com/acoustic-and-soundproofing-products-vocal-booths/soundproofing-materials/limp-mass-barrier-mass-loaded-vinyl-for-vocal-booth/6-17
or a Mass Loaded Vinyl with Acoustic foam (fire retardant) : http://www.vocalboothtogo.com/acoustic-and-soundproofing-products-vocal-booths/soundproofing-materials/advanced-soundproofing-and-sound-absorption-panels/6-18
3 - when you hang the curtains you can hang them from BOTH sides of the passage wall this will have two-side effect - it will double the sound blocking mass and it will create layered barrier with layers of different density. 4 - make sure that your curtain overlaps the opening and creates a tight (as much as you can) seal around the edges. if you can see a light through your cover - the sound will leak through it.
5 - and lastly I am not sure looking at your drawings if the refrigerator sits just in one corner of the room, but is you can you can enclose the refrigerator itself (just on the side facing your room with sound absorbing or sound blocking materials. Do not enclose the whole refrigerator – in needs air exchange to function But if you put another barrier in the direction of your room – it will help.
Good luck. Jeff.