When you attach something, you physically connect one thing to another. You can also connect things figuratively, with logic.
Ex: Once he had attached the antenna to the portable radio, Joe was able to dial in all of the local stations, and we listened to music while we swam in the river.
Ex: We won't be working alone on this project. Jim Finkle from accounts has been attached to it, so that he can address any possible budget issues ahead of time.
When you want to say that a deal comes with certain specific requirements of one of the parties, you say it comes with strings attached. And if you wanted to indicate that someone has decided to spend all of his time with his girlfriend, you might say he has attached himself to her.
Ex: My Uncle Bobby agreed to let us use his car for the weekend, but there were a few strings attached. For one thing, we have to wash and wax it before we take it back.
Ex: I haven't seen Jethro for more than a month. Ever since he started going out with Mary, it's like he's been attached to her.
An old meaning for the word fast is secure, so when you fasten something, you close or attach it together securely.
Ex: The captain began to speak on the PA system, telling us that we were now cruising at thirty thousand feet but since there was a lot of turbulence, we had to keep our seat belts fastened.
Ex: I handed Leila the necklace, and she admired it for a moment before she fastened it around her neck. It was stunning. If you fasten something up, you are closing or securing it completely.
If you fasten onto someone, you are following them closely, particularly when they don't want you to follow.
Ex: Would you mind fastening this up for me? I can't reach behind my back like that, and with the buttons in the back, this is really hard to do.
Ex: The Senator clearly wanted to be left alone, but the reporter fastened onto him, following him around the party, asking question after question. Eventually, the Senator decided to leave.