。
I'm begging you, please hear me out to the end before you fly off the handle.
Literal
Request is-[because-から] hysterics [obj-を] raise before, talk [topic-は] end-[until-まで] listen-please.
お願いですから (literally 'because this is a request') functions as an emphatic 'please, I'm begging you' preface — stronger than bare ください. ヒス is a casual contraction of ヒステリー ('hysteria'), itself a loan. ヒスを起こす is the fixed collocation 'to throw a fit / have a hysterical outburst.' 話は最後まで聞く — literally 'listen to the talk until the end' — is a set phrase for 'hear me out / let me finish.' The sentence is polite but clearly exasperated.