If you look on a bag, a bracelet, or a necklace, you might see a clasp: a little metal thing with parts that fit into each other, so that you can link and unlink two pieces to tighten or loosen the whole thing.
Clasps can be very simple, with two little magnets that snap together and snap apart. (Source)
But no matter their design, clasps always attach parts together, or close things, nice and tight.
So, the word "clasp" can also mean "to hold onto something closely or tightly." You could clasp your own hands together, or clasp your friend's hand in yours.