The usual explanations I see for what Common is and why there's this one language that everyone seems to share are:
- It's a standardized trade tongue.
- It's the official language of whatever empire/nation/state/culture you're in.
- It's what humans speak, and they get everywhere so everyone else learns it in order to tell them to leave.
I don't particularly like any of those. They don't appeal to me, so here's my explanation: Common is the language of adventurers.
It's a professional language full of specialized jargon oriented around travel, exploration, delving, fighting, supplies, logistics, and contract law. You know it because you're an adventurer. Your colleagues know it because they're adventurers. The people you talk to (innkeepers, merchants, guides, nobles, etc.) know it because they regularly work with adventurers and need to communicate specifics while doing business.
When you're not talking about the job, you don't speak Common. It might have words for whatever the topic of conversation is, but that's for work. You have to maintain your professional boundaries. The random people you encounter during the day don't speak Common to each other or when they go home, they only speak it to you during the course of business. It only seems like everyone speaks Common regularly because your character is an adventurer and has an adventurer's view of the world.
It makes sense because adventuring is an exceptionally hazardous profession. You need fast, clear communication in the life or death situations you run into. There's no room for misunderstandings or hesitation.
You need to be absolutely clear about contracts and exactly what you're being hired to do. (Nobles appreciate it because they don't want the party of people who can probably destabilize their realm to feel cheated, come back, and overthrow them.)
You need to be specific about the gear you purchase to make sure it'll be adequate for your nonstandard needs. (Merchants appreciate it because adventurers are good repeat customers and they really don't want a party of vengeful survivors coming back because their gear failed them in the field.)
So you all speak the Common adventuring language to hash out details and make sure you're all on the same page.