When you fasten something, you close it by means of buttons or a strap, or some other device. If something fastens with buttons or straps, you can close it in this way.

If you fasten one thing to another, you attach the first thing to the second, for example with a piece of string or tape.

If someone or something fastens your attention on a particular thing or if your attention fastens on it, you start to concentrate on it rather than on anything else.

If someone or something fastens on a particular thing, they start to concentrate on it.

If someone fastens on you, they keep following, talking to, or staying with you, when you want them to go away.