Why is the fgets function deprecated?

No, fgets is not actually deprecated in C99 or the current standard, C11. But the author of that tutorial is right that fgets will not stop when it encounters a NUL, and has no mechanism for reporting its reading of such a character.

The fgets function reads at most one less than the number of characters specified by n from the stream pointed to by stream into the array pointed to by s. No additional characters are read after a new-line character (which is retained) or after end-of-file.

(§7.21.7.2)

GNU's getdelim and getline have been standardized in POSIX 2008, so if you're targeting a POSIX platform, then it might not be a bad idea to use those instead.

EDIT I thought there was absolutely no safe way to use fgets in the face of NUL characters, but R.. (see comments) pointed out there is:

char buf[256];

memset(buf, '\n', sizeof(buf));  // fgets will never write a newline
fgets(buf, sizeof(buf), fp);

Now look for the last non-\n character in buf. I wouldn't actually recommend this kludge, though.

This is just GNU propaganda. In no official sense is fgets deprecated. gets however is dangerous and deprecated.