Considering the magnification effect of APS-C sensors, they theoretically have to extend the zoom range of telephoto lenses. Accordingly, would it be wiser to use an APS-C camera for shooting wildlife instead of full frame, or this is just something that I only imagine and it'd be different in practice?
The APS-C sensor merely crops the image that would have been captured on a full frame sensor, so you end up with what you'd get if you used a full frame and cropped in post (see: Does my crop sensor camera actually turn my lenses into a longer focal length? and Is crop-factor a bad thing?)
But given a full-frame and a crop sensor of the same resolution, say 14MP, at least with the APS-C you'll have a 14MP end result. With full-frame, if you crop the image to match the APS-C image, you'd throw away pixels and end up with a 6MP image.
Crop sensors are indeed used for wildlife to get more reach without sacrificing megapixels. And, you can get closer images without spending as much money. Sure, you could crop, but then your printing dimensions will be reduced. For display on the web, at 72dpi or so, it wouldn't matter if you cropped.
All that said, remember that to get the same image as a full frame sensor, you have to apply the crop factor to both the focal length and the aperture. Many photographers forget this, and think that their full-frame 200mm f2.8 lens will turn into a 300mm f2.8 on a crop sensor. Actually, you get the effect of a 300mm f4, instead.
There can definitely be some benefit to be gained by using a crop sensor camera when longer focal length is desireable. It is one of the reasons compact "superzoom" cameras can give fields of view equivalent to 1000mm+ focal lengths on a full frame camera with a much smaller lens than would be required to get that same FoV using a full frame sensor camera.
If the pixel density is the same on a FF camera and an APS-C camera, then there would be little to no benefit (all other things being equal). But the pixel density is rarely the same. A typical APS-C camera might have a 20MP sensor that gives a pixel spacing of about 4 microns. A 22MP FF camera has a pixel spacing of 6.25 microns.
The crop factor of a particular sensor is based on linear measurements, but pixel density is measured in terms of pixels per area. This means the number of pixels on a FF sensor that fall within the space covered by a 1.5X APS-C sensor is not 1/1.5X, but 1/2.25X of the total number of pixels on the FF sensor. In the case of our typical 22MP FF camera, if we crop the image by 1.5X in the linear measurements we are only left with a pixel count of just under 10MP. In the case of cropping to match a 1.6X APS-C sensor it would only be about 8.6MP.
If 8-10MP is sufficient for your intended display size, then there will be very little difference between cropping the FF image or using the APS-C camera (all other things being equal). But if you need more than 8-10MP resolution for your intended display size, then the advantage of the APS-C body becomes obvious.
If you magnify the image projected by the lens, such as is the case with a teleconverter, you spread the light thinner and give up aperture in terms of the f-number used to calculate exposure. But if you increase magnification when printing or displaying the image you don't lose any aperture in terms of exposure. You do, however, lose performance in terms of perceived image noise when you magnify an image. Displaying the same image at different viewing sizes (magnifications) also affects the depth of field as perceived by the viewer at the same viewing distance.
Beyond the relative pixel densities, there is also the issue of comparing different generations of technology. A newer FF body might do better cropped to APS-C magnification compared to an older APS-C body. On the other hand, a newer APS-C body may do better when compared to a cropped image from an older FF body.
Also please note that when you crop a FF image to the same magnification as an APS-C sized image, you have just thrown away the extra light that the FF sensor collected to give it that one stop advantage in terms of noise.
There's no universal answer. Cameras are different, with differing resolutions, pixel pitch, and noise performance. A newer camera can trump a bigger camera. Does the bigger one have bigger pixels, more pixels the same pitch, or something in between, which makes them harder to compare?
Also, you might be able to afford a much better lens (or only use the center part thus making a cheaper lens perform better) with a smaller sensor. Is the better optics a bigger or smaller effect than the difference in noise?
Compare actual cameras and equipment that would make up the kit, not one single property held by many cameras of different brands and tiers. I've had a Sony alpha 6000 (crop) take a better picture than a Canon 5D (full) in the identical shot: the larger sensor gives a 1 stop advantage to low-light. The other camera had its own advantages. One stop due to one feature doesn't just trump everything else that's going on.
Full frame cameras will give you better quality images especially in low light. However, they are bulky, heavy and expensive. And yes you do lose telephoto capability to a degree as the crop factor is not there. So you will need a bigger lens on a full frame to cover what you lost with a crop sensor camera. Which ofcourse will add to the weight. Some people are willing to live with this headache to get the best images possible. But others are not. Using cameras in real life in the field is always better teacher than theorizing about it. I currently use a crop sensor DSLR for nature photography and carry only 2 lightweight lenses. Since weight is an issue for me when i travel across the world. I have used all kinds of cameras on my travels and i am always looking to get the right balance between weight, bulk and performance.
Most answers here compare the use of a crop-sensor camera with the option of cropping in post. This raises the question of comparing either megapixel counts or pixel densities, as discussed in all the answers.
However, all these answers fail to address a very essential issue: viewfinder magnification. Crop-sensor cameras almost universally have a larger viewfinder magnification than full frame cameras, which is only natural given their smaller focusing screen. If you are going to use only an APS-C (or smaller) area of your sensor anyway, an APS-C camera will give you a better experience in terms of composing your shot in the finder. With full frame you will have to compose on a smallish partial viewfinder image.