To let loose, cut loose, or behave exuberantly and without restraint; to have a wild/giddy time (colloquial).
学園祭でみんながはっちゃけて踊っていた。
At the school festival, everyone was letting loose and dancing.
飲み会で彼ははっちゃけて、普段見せない一面を見せた。
At the drinking party he cut loose and showed a side of himself he usually doesn't show.
週末だから今日は思い切りはっちゃけよう。
It's the weekend, so let's go all out and have fun today.