The recognition that in order to achieve one's purposes, one must act oneself ("take" responsibility) rather than expecting others to do something (compare initiative).
The etymology of the word ultimately relates to Latin respondere (to reply).
Generations of worry about a ChristianLast Judgement have engrained a sense of "personal responsibility" into ethical decision-making. Any action seen as causing a situation can (in a hyper-critical society) become an occasion for sheeting home "responsibility" (blame).
In politics, calls for responsibility often form a pejorative means of attacking political opponents. This habit of demanding behaviour aligned to one's own desires also occurs in other arenas: one expects "responsibility" from children, spouses, workmates and employees, meaning they should change their attitudes to suit the speaker.