어 제끼다

do it flat out, with abandon

Slangy energy tacked onto a verb: whatever it is, you do it hard and without holding back. Every line here is one of two phrases — 떠들어 제끼다, talking or chattering away at full tilt ([특강답게 알차게 떠들어제껴볼게요]), and 흔들어 제끼다, dancing or shaking it (여러분, 흔들어 제껴, 월요일이에요). 제끼다 is the casual spelling of 제치다, and the whole thing belongs to hyped-up spoken language and captions.

Register: very casual, upbeat slang