drywall
I bought a home recently which was built in 2004.
I installed a TV mount on a stud and I am now trying to run cable behind drywall, but when I cut through my drywall I found wood sheet behind my drywall. What is that for? Can I cut through that wood?
As shown in the picture, the TV is mounted on a stud and stud finder is also attached to that stud for reference. 
the thickness of a single sheet of drywall
Did they happen to specify how thick that single sheet should be? I bet that they didn't and are being over protective, because they sell drywall in varying thicknesses from 1/8" up to 5/8".
But I don't think they're being TOO over protective, because I think that's the exact same mount I used, and it mounts using lag bolts. Lag bolts are very heavy duty bolts that are driven into the wood and bite into it to create a very strong anchor.
Their strength, however, is in the depth to which they sink in the underlying wood - the deeper they sink the more thread length is biting into the wood holding on tight. So - if you get too shallow of an installation, you don't get the strength you might need.
I read another answer where someone claimed they had a 50" TV mount just in drywall, no studs!
Not for long, I bet...
The other option I considered (although am loathe to attempt) is to cut away the extra drywall, attach some more wood onto the stud (enough to make up why width of the drywall I removed) and attach the mount to that.
You could do that - I thought about doing the same thing - and the way I would do that is to use a multi-tool (below) to cut out the drywall in the same shape as the back plate of the mount, affix that to the studs very securely, mark the stud location on it and drill through the plate into the studs and mount the panel.
HOWEVER - I suspect there's an even simpler solution.
Get longer lags.
They come in a wide assortment of depths and diameters. Just take the lag bolts they supplied with the mount, go to Lowes/Home Depot - and find one that's the same diameter as what they supplied and about 1" longer than what they gave you, and use those instead.
Cut 3/4 inches over each stud. (Oscillating multitool is the best for this, but patience and a utility knife will suffice.)
Screw new drywall on. You could use a thin bit of plywood at the top horizontal seam to stiffen it up. Screw the old drywall as well. (Be careful about the power wires.) Tape and skim the seams. Caulk the joint between the baseboard and drywall. (Texture? I can't tell from the pic.) Paint.
Best Answer
It’s undoubtedly a shear wall to resist lateral loads, especially if you live in a “high wind” area or a seismically active area.
It could also be a reinforced wall for shelving.
Yes, you can cut through it, or cut openings in it, whether it’s a shear wall or not. I would not make an opening more than 1/4 the height of the wall and not more than square shaped...don’t make it a large rectangle. Also, I’d install framing members at the edge of the opening. (If it’s a load bearing wall, you’ll need to install a header.)