According to Nikon, the D5200 has a viewfinder magnification of

Approx. 0.78x (50 mm f/1.4 lens at infinity, -1.0 m-1)

This means a magnification such that the image of an object you see when looking in the viewfinder would be the same size as looking at that same object with your naked eye would be with an approximately 64mm focal length. To get the same 8X magnification you get with your 8x42 binoculars you would need a 513mm lens.

With a 300mm lens you would get roughly 4.7X magnification at the viewfinder compared to looking at the same objects at the same distance with your naked eye.

Note that with a 1.5X crop body, the diagonal angle of view rendered with a 513mm lens would be around 3.5°, so to get the same magnification factor as you look through the viewfinder, you'd give up quit a bit of field of view. This is because the exit pupil of your D5200's viewfinder is smaller than the exit pupil of your 8x42 binoculars with a 6.3° AoV. That 300mm lens, on the other hand, would give you about 5.5° diagonal angle of view.

In other words, the size of the circle you see in your binoculars would have almost twice the diameter as the diagonal of the rectangle you see in your camera's viewfinder. (This assumes the eye relief, or the distance your pupil is placed behind the exit pupil, is the same for both.)

For more about how focal length, sensor/focusing screen size, and viewfinder magnification interact to affect the size of objects when viewed through a viewfinder, please see this answer to Can I convert binocular zoom to equivalent lens focal length?