The appearance of an "arch" is a distortion caused by stretching the
image to fit on your flat screen. It is not real.
I have seen a number of explanations of this that claim it is due to lens distortions or projections on the flat plane of the image/screen. I don't think either one is correct. If you are ever in a dark-enough place to actually see the MW with your own eyes, it looks like an arc any time the band is lower than 45 degrees or so above the horizon. If you take an image of the MW at that point with a strictly rectilinear lens, mounted with nodal slide to conrtrol for paralax, and then stitch it together it will still look like an arc. Yes, the arc can be exaggerated by lens distortion, but that is not the primary reason the MW looks like an arc.
A number of apps that predict the position of the MW exist today, and they clearly show the COMPUTED position of the MW in the sky as an arch (see for example PhotoPills or Stellarium). The appearance of an arch is due to the projection of the MW on the curved sky, not on "your flat screen". The solid angle subtended by the MW is very small, and when you view it from a point on the Earth's surface that is substantially out of the projection plane it looks like a curve. As pointed out above, when the MW projection plane directly intersects a large circle of the Earth AT THE POINT WHERE YOU ARE STANDING the band is directly overhead and it looks like a flat plane.
The closest analogy I can come up with is to think of a LARGE hoop. Place your head in the center of the hoop and look at it edge-on. It looks like a straigt line. Now rotate the hoop slowly around an inplane axis while maintaining your head in the center. It starts to look like an arc.