The words "their king" is likely a reference to God. Possible meanings are 1) "rejoice because he is their king" or 2) "rejoice because their king is good."
# praise his name
The word "name" is a metonym for Yahweh himself. AT: "praise him" or "tell people how great he is" (See: [[rc://en/ta/man/translate/figs-metonymy]])
# tambourine
a musical instrument with a head like a drum that can be hit and with pieces of metal around the side that sound when the instrument is shaken (See: [[rc://en/ta/man/translate/translate-unknown]])