Magic numbers are values, not generally visible unless you dump file contents using a command like od that displays file contents in hex, that serve as signatures for file type identification. Some are fairly obvious once you see them -- like if you spot "\x89PNG" in a file dump. Others won't offer much of a clue.
How are they used?
The tool for file identification is a command named "file". Use the file command to examine your oddly named my.image file and might just tell you something like this:
$ file my.image
my.image: JPEG image data, JFIF standard 1.01
The "JFIF" tag in this description stands for JPEG File Interchange Format. This output tells you that this complies. This tells you that the magic number that is associated with JPEG files is both stored in this file and stored in the right location to identify the file. The same value at some other location would have no effect and would likely be coincidental.
Magic numbers help ensure that Unix systems have more to go on than file names when trying to identify file content. Examine the /usr/share/file/magic on your Linux system and you'll notice that there's a lot of information there about different types of files.
