A bracelet is held together by a clasp. A girl who gets a nice one from her boyfriend might clasp her arms around him. A clasp is a fastener. To clasp is to hold tightly.
In all uses of the word, clasp means to hold together tightly. You want your bracelet or belt clasp to be strong so it doesn't come apart. And when you take a child on a walk across a busy intersection, you clasp their hand tightly. The word is not related to the word clap, but if you clap your hands together, then keep them there, you turn a clap to a clasp.