Attachment-of the verb to attach-is synonymous with clinging, tying and even establishing dependence or obsession with something or someone.
When you get attached to an object, idea, activity or person, you end up generating strong links with them. So strong that you can become addicted to them.
And, when you're addicted to something, your life ends up turning around that addiction. You are willing to do everything to not let go of that object, activity or person. You do everything in your power to stay with them and nurture the bonds that bind you.