Praise be to Allah.
Waswas is a kind a sickness, may Allah keep us and you safe from it. It is caused by the shaytan gaining control over a person and filling him with anxiety, grief, distress and worry.
Waswas is of various kinds, such as waswasah about divorce, where the person affected thinks that he has divorced his wife, or thinks to himself about divorce and he is uncertain as to whether he has uttered the words of divorce or not, and he remains confused, although he has no intention or desire to divorce his wife.
Some of them think that if they say a certain word to their wives or they do not respond to them, that divorce has taken place thereby, and that this is a metaphor for divorce. All of that is an illusion that has no reality. Perhaps he feels that the words came out of his mouth and he heard them. Even if we assume that such a person did speak audible words, then, as long as he is affected by waswas and did not really intend a divorce, then his divorce does not count. This is the meaning of the words of al-Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) quoted in the question.
Some people experience waswas with regard to purification; every time they do wudu, (ablution) they repeat it, thinking that they did not wash one part of the body, or that their wudu is lacking, or that if they do not repeat wudu they will remain uncertain for the rest of the day. Thus he is overwhelmed with a strong urge to repeat his wudu, and he will not feel at ease until he does so. We have mentioned in a number of answers that the way to treat waswas is to strengthen one’s faith by doing righteous deeds and reciting beneficial dhikrs (words of remembrance), and also not paying attention to the waswas or responding to it.
And Allah knows best.