It looks like this video was taken somewhere in the middle latitudes. I can't tell exactly where, but if it was taken near the equator or close to the poles, it would look quite different.* For this answer, I'll just assume it was taken somewhere around 45° north.
(We can tell it's not 45° south because then the Milky Way and the surrounding constellations would be upside down compared to what's seen in the video.)
Anyway, at 45° north, the north celestial pole is located due north (obviously!) at 45° above the horizon. (If you know how to find Polaris in the night sky, you can easily pinpoint its location to within less than a degree.) As the Earth rotates around its axis, everything in the sky will appear to move in concentric circles around the celestial pole.
(Well, everything except satellites, meteors, airplanes, auroras, clouds and other stuff close to the Earth, at least. The Sun, the Moon and the planets will also deviate slightly from circular paths due to their proper motion, but not enough to make a significant difference in just one day. But for the fixed stars, including the Milky Way, the circular path around the celestial pole is exact, at least on human timescales.)
Here's a long exposure photo of star trails around the north celestial pole from Wikimedia Commons (photographed by 0x010C, used under the CC BY-SA 4.0 license) to illustrate this:
Anyway, let's imagine that we could see the lines of celestial latitude and longitude as a giant map grid drawn in the sky — something like the picture below (public domain, from freesvg.org) but viewed from the inside, from a point near the center of the sphere:
The point(s) where all the lines of longitude meet are the celestial pole(s), and the star trails in the earlier photo above would run parallel to the lines of latitude circling the poles.
Now, standing on the surface of the Earth, with this imaginary sky grid around us, we can only see (about) half of it at any given time — the other half is below the horizon. But the grid is rotating around the celestial pole, so parts of it continually disappear on the western horizon as other parts rise into view from the east.
Near the pole, the lines of latitude will form complete circles in the sky around the pole, and stars at those latitudes will never set below the horizon. But the latitude lines further away from the pole will be partly hidden below the horizon, so that only a part of the circle will be visible as an arc in the sky, which stars close to it will follow from the eastern horizon to the western one.
Now imagine that you're looking due south, at a point where one of the lines of longitude crosses the equator at its highest point above the horizon. Obviously the equator will appear parallel to the horizon there (since every arc does, at its highest point!), and the line of longitude, crossing it at a right angle, will look perfectly vertical.
(Or you could just imagine looking at the actual night sky, and at some feature that crosses the equator at an approximately right angle — like, say, the Milky Way.**)
But now let's wait a bit, keeping our attention on the imaginary crossing of these two imaginary lines, as the celestial sphere rotates around us and the crossing point moves along the arc of the equator in the sky. Clearly, as we follow the arc of the equator from its highest point down towards the horizon, it cannot remain horizontal, but must turn downwards. And, conversely, the lines of longitude crossing the equator also cannot remain vertical but must turn along with the equator to maintain a right angle with it.
At the horizon, the celestial equator will cross it at an angle that equals 90° minus the latitude you're standing at on the Earth's surface. And, conversely, when the crossing point we were tracking with our mind's eye meets the horizon, the line of longitude (which was originally vertical at the apex of the arc), still being at right angles from the equator by definition, must be tilted at an angle equal to our latitude on the Earth's surface.
*) As I noted at the beginning, things would look different near the poles or close to the equator.
Near the north or south pole, of course, the corresponding celestial pole will be almost directly overhead, and stars will move in circles around it, approximately parallel to the horizon at all times. Only stars very close to the horizon will ever rise or set, and lines of longitude will always remain approximately vertical, so you can't see this phenomenon there.
Conversely, at the equator the celestial poles will be at the horizon, and the celestial equator will pass directly overhead. Lines of longitude will be parallel to the horizon where they cross the equator, except directly overhead at the zenith where "horizontal" and "vertical" become meaningless. (In technical terms, the observer-relative spherical coordinate system has a singularity there.) So anything in the sky near the celestial equator will appear to rise straight up from the east, without rotating relative to the horizon, until it crosses the zenith and starts descending to the west, now upside down.
This is perhaps most notable visible with the crescent Moon which, as seen from near the equator, will appear to rise with the "points" of the crescent pointing straight up (or down) and to set with the points straight down (or up).
**) The Milky Way actually crosses the equator at about a 63° angle, not 90°, but eh... it's close enough, and the exact angle really doesn't matter much here. It will still (appear to) rotate along with the celestial coordinate grid near it.
Also, in the video in your question, the camera does not seem to point exactly at the equator, but rather appears to be pointed at the galactic center, located at about 29° south of the celestial equator. Qualitatively, though, that also makes fairly little difference. FWIW, the actual celestial equator seem to run close to the top of the picture.