Essentially, yes. If you used a full frame camera and an 810mm lens you would get about the same field of view as what you get with an ASI183MC and a 300mm lens. It doesn't actually make the 300mm lens an 810mm lens. It just changes your filed of view. It's referred to as "crop factor" because it's as if you cropped a smaller portion of the image you would normally get from a full frame sensor with the same lens. If you took an image with the full frame sensor and the 300mm lens, your field of view would be about 412 x 275 arc-minutes. If you took an image of the same target, centered on the same thing, using the ASI183MC and the 300mm lens, with it's 151 x 100 arc-minute field of view, that image would essentially be the same as if you cropped out the center 151 x 100 arc-minute portion of the full frame image. So while it looks like you've used a longer lens, in reality you are just recording a smaller portion of the available image circle the lens produces. But the crop factor is useful for helping us get an idea of what the field of view will look like, so in general use we tend to look at it in terms of an 810mm vs a 300mm lens.
