It means the father is walking forward toward another person, and making "Ssss" sounds. It most likely means the father's mood is very threatening and unpleasant.

Andi's answer is correct. I would like to add: 1) A Google search shows me that this is a sentence from an English translation of Franz Kafka's story, "The Metamorphosis." 2) This is a colorful piece of very original creative writing. It is not everyday English. 3) A few paragraphs later, we read: "If only his father had not hissed so unbearably." "Pushing out sibilants" is a colorful way of saying "hissing." 4) It is a translation, and maybe the translator is using an unusual phrase to capture something about the phrasing in the original German.