The phrase "I know my way around [something]", "He knows his way around [something]," etc., is a common phrase at least in the USA. The person who knows their way around [something] is a person very experienced in [something]. It means the person knows many aspects of that something, whatever it might be. I know my way around Tucson, Arizona (my city): Means I'm very familiar with the roads of Tucson and can probably get from most points to most other points within the city without a map. This version of "I know my way around..." is probably the first, referring to somebody knowing a physical area well. I know my way around a guitar: Means I know how to play guitar very well. I know my way around a computer: Means I know how to use computers very well. I know my way around tiny things... who knows ... I think you need more context to figure out exactly what was meant in that case. It might have been a sarcastic statement when taken in context.

It means to be very familiar with something, maybe a place or maybe an activity. For example, he knows his way around that big house, or he knows his way around the park, but also she knows her way around the guitar, or she knows her way around the computer industry.