How do they make the sun or moon appear so big in backgrounds of some images?
As per Era's link in the comments above, photographers have been fooling people with trick photography for over a hundred years now, and your included photo is a classic example.
If you see a photo that doesn't seem quite right for any reason, it probably isn't.
It is entirely possible to create a larger-than-life image background like this in camera by using a very long lens and taking advantage of the tele compression effect. To do this, you need your foreground subject (the person and tree in this case) to be just the right distance from the camera so that their relative size compared to the background (moon) is what you want. The formula for the camera-subject distance depends on the focal length of the lens, but the constant factor is the moon - it will always be the same size in the frame, since you can't get (significantly) closer or further from it.
In other words, if the subject is too far away from the camera, their relative size will be too small and they will appear as a small speck in front of the moon; too close and they will appear larger than the moon and more 'normal'.
EDIT: Speaking of being fooled, I was mistaken in my initial answer; upon closer inspection this image does indeed appear to be taken in camera, as opposed to being collaged later.
however, your example image appears to use the common technique of collaging two or more images to achieve much the same effect. In this case an image of the moon has apparently had a silohuetted image of a man and tree superimposed in front of it, probably using Photoshop or a similar editor.
How do I know this? I don't, but if you spend enough time editing digital photos you can sometimes see the telltale signs of manipulation. In this case the two things that strike me as unlikely to be natural are:
- The moon shows no sign of atmospheric haze, as would be visible in most parts of the world with the moon near the horizon (assuming the moon is near the horizon).
- The tree leaves block all of the bright moonlight except for a few tiny spots. Although it's possible the foliage is dense enough to do this, it is unlikely in my opinion.
These observations are not a criticism, and for all we know the silhouette could be a complete fantasy - drawn by hand. In which case, kudos to the photographer; this is a great image.