Here is the fragment again:
Hissing and spitting like an angry alligator caught by its tail.
You correctly identified the fragment as a participle phrase. Hissing, the present participle that begins the phrase, is the clue.
To fix this fragment, you could add a subject and verb, like this:
The old coffee maker was
hissing and spitting like an angry alligator caught by
its tail.
Or you could use Punctuation Rule 3:
Main Clause + , + Participle Phrase.
The correction would look like this:
The old coffee maker brewed Christine's breakfast,
hissing and spitting like an angry alligator caught by
its tail.
Or you could use Punctuation Rule 4:
Participle Phrase + , + Main Clause.
And the correction would look like this:
Hissing and spitting like an angry alligator caught by
its tail,
the old coffee maker brewed Christine's breakfast.
Enjoy the honor!