IPA: /ˌæpəˈplɛktɪkli/
KK: /æpəˈplɛktɪkli/
In a way that shows extreme anger or rage, often to the point of being unable to speak or act normally.
He reacted apoplectically when he heard the news about the unfair decision.
Apoplectically is derived from "apoplexia" (from Greek "apoplexia", meaning a stroke or paralysis) and the suffix "-ally" (meaning in a manner). The word describes behaving in a manner that resembles or is characteristic of a stroke, often indicating extreme anger or rage.
Think of someone acting in a way that resembles a 'stroke' ('apoplexia') — that's why apoplectically means behaving in a manner of extreme anger.