Definition
Data roaming is a feature that lets your phone use the internet abroad by borrowing a local carrier's network instead of your home carrier's. Because the two carriers made a deal ahead of time, you can connect to the local network without swapping your SIM card. The local carrier bills your home carrier for what you use, and your home carrier adds it to your bill.
Why your library card works at another library
Say you go to the library at a neighboring university instead of your own. If the two schools have a partnership, your student ID alone lets you check out a book on the spot. You don't need a new membership card.
Your phone connects abroad in a similar way. You step off the plane and turn off airplane mode, and the phone scans the airwaves around it. Your home carrier's towers are nowhere to be found. Only the local carrier's towers show up.
The SIM card inside your phone plays the role of the student ID. It records which carrier, in which country, you subscribe to. The local carrier passes that information back to your home carrier to check that you're a real customer.
Once that's confirmed, the local network makes room for you. Keeping your own number and plan while riding on someone else's network is called roaming, and the part that covers internet use is called data roaming.
How do the charges get added?
Say you return a book late at the neighboring library. You don't pay the late fee right there. The neighboring school tells your own school, and your school collects it from you.
Roaming charges follow the same path. The local carrier records how much data you used and bills your home carrier. Your home carrier adds its own cut and folds the total into your bill for the next month.
Because an extra company sits in the middle and has to settle up, prices are often higher than at home. In the past, it was common for people to be shocked by a huge bill after just a few days of travel.
That's why many carriers now offer roaming plans with a set price per day. Also, if you turn the data roaming switch off, only the internet is blocked while calls and texts still work. But since calls and texts keep working, roaming charges can still apply to calls you make or receive and texts you send.
A few more details
Wi-Fi roaming, which sounds similar, is a different thing. It keeps your connection from dropping as you move between routers inside a building. It has nothing to do with crossing borders and borrowing carrier networks.
You can also use a local carrier's SIM card, or a travel data SIM or eSIM. That's closer to getting a brand-new membership card at the neighboring library. It's often cheaper than roaming, but receiving calls and texts on your own number gets harder. Some travel products even work by having a carrier from another country borrow the local network.
An eSIM is a type of SIM where the line information is downloaded into a chip built into your phone. So you can use your home line as an eSIM and roam with it, or keep two cards by pairing it with a regular SIM. Choose based on what matters more to you on your trip.
Either way, the basic idea is the same. It's just a choice between using your own student ID as is, or making a new card for that school.
Common misconceptions
If data roaming is on but I'm not using the internet, I won't be charged.
Even when you aren't looking at the screen, apps fetch notifications and content on their own in the background. That data counts toward your bill too. Turning the data roaming switch off blocks this data, but roaming charges can still apply to calls and texts.
Roaming works because my carrier has set up towers overseas.
Your carrier doesn't have equipment abroad. You borrow a network that the local carrier has already built, and the carriers settle the cost between themselves later.
Where you meet it
The moment you land at an airport abroad and turn off airplane mode, a local carrier's name and a welcome text pop up on your phone.
After you use a map app to find your way while traveling, the cost of that data is added to your regular bill at home the next month.
In one sentence
Abroad, you keep your own number and SIM while borrowing a local carrier's network. The carriers settle the cost between themselves, and it ends up on your bill.