We can see it because it refects the light of the sun. It doesn't reflect much light (most comets are dark grey to black on their surface), but as you mention, it is exceptionally large (for a comet). Like most other solar system bodies, it does not glow at visible wavelengths. We see only the reflected light of the sun.
So it initially appeared as a star-like dot in photographs taken with a powerful telescope. The dot was very faint, which is why only a large telescope could detect it. More photographs established that this dot slowly moved relative to the stars and it was, moreover, found to have been on some older photographs, but not noticed at the time.
Once you can see how the dot moves, you can work out its orbit, including its long period, and its slow, distant pass of the sun.
Later photographs show a weak "coma", formed of gas released from the surface of the comet, even at a great distance from the sun.
This comet is unusual. Typically, long period comets are much smaller, and only noticed when they have fallen much closer to the sun, heated up and produced a sizable coma. As they have fallen further, they are moving fast, and travel quickly through the inner solar system.
This comet is large, but it won't enter the inner solar system. Instead, it will travel slowly through the outer solar system. Since it remains far from the sun, it will be visible in professional telescopes for a long time, but it will never get bright enough to be seen with the naked eye.