Shouldn't it just relate to the magnifying power of a mirror (in reflecting telescopes) or lens (in refracting telescopes)?

Apparently you don't understand how a telescope works. There isn't a single "magnifying power of a mirror ... or lens." A simple telescope has at least 3 parts: an objective optical element (closest to the observed object), an eyepiece optical element, a "structure" for maintaining the geometry of the objective and eyepiece.

The objective, usually with a fairly long focal length, produces a potentially-real image in the region near the eyepiece. The objective can be a mirror or a lens, hence the distinction between reflecting and * refracting* telescopes, respectively.

If the eyepiece is a positive focal length lens, the image should be produced between the objective and the eyepiece. If the eyepiece is a negative focal length lens, the image should be slightly behind the eyepiece. This means that the optical path distance between the objective and the eyepiece must be close to the focal length of the objective.

The size of the image produced by the objective is much smaller than the original object viewed! This is counter-intuitive. On the other hand, the image is much closer to your eye, so it occupies a larger angular space in front of your eye, hence, the angular magnification. For an example, hold a penny in your fingers at an arms length from your eye. Then bring it closer to your eye. The penny image appears larger, but it really isn't. The eyepiece acts both as a magnifying glass to expand the image and, more importantly, a focusing device so that your eye lens can produce a sharp image on your retina. The angular magnification is $$M_{\alpha}=-\frac{f_o}{f_e}$$ where $f_o$ is the focal length of the objective, $f_e$ of the eyepiece, and a negative value of $M_{\alpha}$ means the image is upside-down.

With a reflecting telescope, the optical path easily can be folded, so a longer focal length can be used in the same physical length. That means that it is possible to get a larger angular magnification system in the same physical length compared to the refractor.

With larger angular magnification, a small real angle will occupy the same visual field as a medium real angle at small angular magnification. For example, if the width of the visual field of the user through the eyepiece is 10 degrees, then a 40X telescope system will have a 0.25 degree width field of view, whereas a 100X telescope system will only have 0.10 degree width field of view.

There are other things, such as objective and eyepiece diameters, which can reduce the actual field of view below the ideal, but the ideal is limited directly by $M_{\alpha}$.

Shouldn't it just relate to the magnifying power of a mirror (in reflecting telescopes) or lens (in refracting telescopes)?

There isn't a single "magnifying power of a mirror ... or lens." A simple telescope has at least 3 parts: an objective optical element (closest to the observed object), an eyepiece optical element, a "structure" for maintaining the geometry of the objective and eyepiece.

The objective, usually with a fairly long focal length, produces a potentially-real image in the region near the eyepiece. The objective can be a mirror or a lens, hence the distinction between reflecting and * refracting* telescopes, respectively.

If the eyepiece is a positive focal length lens, the image should be produced between the objective and the eyepiece. If the eyepiece is a negative focal length lens, the image should be slightly behind the eyepiece. This means that the optical path distance between the objective and the eyepiece must be close to the focal length of the objective.

The size of the image produced by the objective is much smaller than the original object viewed! This is counter-intuitive. On the other hand, the image is much closer to your eye, so it occupies a larger angular space in front of your eye, hence, the angular magnification. For an example, hold a penny in your fingers at an arms length from your eye. Then bring it closer to your eye. The penny image appears larger, but it really isn't. The eyepiece acts both as a magnifying glass to expand the image and, more importantly, a focusing device so that your eye lens can produce a sharp image on your retina. The angular magnification is $$M_{\alpha}=-\frac{f_o}{f_e}$$ where $f_o$ is the focal length of the objective, $f_e$ of the eyepiece, and a negative value of $M_{\alpha}$ means the image is upside-down.

With a reflecting telescope, the optical path easily can be folded, so a longer focal length can be used in the same physical length. That means that it is possible to get a larger angular magnification system in the same physical length compared to the refractor.

With larger angular magnification, a small real angle will occupy the same visual field as a medium real angle at small angular magnification. For example, if the width of the visual field of the user through the eyepiece is 10 degrees, then a 40X telescope system will have a 0.25 degree width field of view, whereas a 100X telescope system will only have 0.10 degree width field of view.

There are other things, such as objective and eyepiece diameters, which can reduce the actual field of view below the ideal, but the ideal is limited directly by $M_{\alpha}$.