It's actually pretty simple. Diseases such as macular degeneration and diabetic retinopathy result in central scotomas, or blind spots, within the field of vision of the sufferer. When viewing an object, these blind spots "obstruct" the object, partially or completely, making the recognition of the object difficult or impossible. If the object is magnified, however, the blind spots are less of an obstruction, making the object's recognition and identification much easier.
The photos below help to illustrate this concept. On the left is the simulated view of an excerpt of print through the eyes of someone with macular degeneration. On the right is the simulated view through the eyes of the same sufferer but after employing 4 times magnification.
Notice how the blind spot on the left obscures many entire letters. On the right, however, only parts of the letters are blocked, making it possible to identify and read the words.

Magnification can be provided in different ways. The object itself can be made larger, such as large-print text in a book or greeting card, the viewer can get closer to the object, or an optical or video system can be used.
Visit Types of Magnifiers to learn about the many options available in magnifying devices.