'Stumble' is to walk in an uncontrolled manner, such as when you are drunk or drugged. If you stumble, you may fall over, or almost fall. 'Stumble out' is not a phrase in itself. However, if you want to say that someone left a particular place in a stumbling manner, you could say, for example 'We stumbled out of the pub at midnight'.

It's not a special phrase, at least in US English. It just means to "stumble" or "have a small problem when walking" while going "out" from someplace. "He stumbled out of the bar."