Something that's fastened is securely attached or closed, like the fastened latch on your front door or the fastened buttons on your jacket.
The adjective fastened and related verb fasten come from the Old English fæstnian, "make fast or firm, fix, or secure." When you snap your raincoat, you fasten it — and you can then describe it as fastened. Anything that's closed or held in place this way is also fastened, whether it's a gate, a necklace, or a seat belt.