。
I got down on my knees and begged that girl.
Literal
Kneel-[て form] that girl [locative-に] asked [emphatic-よ].
A dramatic gesture of begging. 跪く (ひざまずく) literally 'to bend the knee' is 'to kneel'. The て-form links the kneeling to the asking as a sequential action. あの娘 with the kanji 娘 read as 'ko' (reading non-standard for 娘 which is normally むすめ/こ) is a slightly archaic/literary way to refer to 'that girl', common in song lyrics and period dramas.