竟敢 combines 竟 (unexpectedly/actually) with 敢 (dare), giving the sense of 'actually dare to' or 'have the audacity to'. It's used when the speaker is surprised, disapproving, or impressed by someone's audacity. You can use it in both positive and negative contexts, but it usually carries a tone of reproach or amazement. For example, 他竟敢这样做 means 'He actually dared to do this (and I'm shocked/displeased).' The negative form 竟敢不 means 'dare not to' in a context where you expect someone to comply.