I think that most people would interprete "to level up" with "progressing to a higher level or standard", usually a step upwards through learning, studying, practising, exercising, gaming, challenging, improving one's methods, or defeating an adversary, gaining, collecting or winning points, awards, etc., in one or several specific domains (in this case in computer games - I am no expert in this regard).
It refers also to optimising, improving, increasing, adding, easing, or adapting things, systems, capacities, situations or spiritual considerations in various real life domains and certainly also in fiction.
But after these rather awkward definitions, I checked the Urban Dictionary, and it has indeed some more definitions:
nerds, for instance, use it for "getting laid" in an ironic sense, there's also a definition about "feeling the affects (or effects?) of alcohol intoxication"; oh, they are also mentioning the ingestion of magic mushrooms and a trip, as a clever play-off in the Super-Mario games and, in the end, (citing):
"The first heavy daze of the day achieved through drub/alcohol substance; to level up is to achieve a higher level of consciousness; in reference to hit points, the term is often followed by a sound effect of video gaming nature".