This word was invented by English comedian Sacha Baron Cohen for his character Ali G in "Da Ali G Show". Since Sacha is Israeli, he pulled from the Hebrew language and urban slang to create Booyakasha. First, the very popular Hebrew phrase "Sheket B’Vakasha!" which means "Please be quiet!". Second, the slang term booya, which is a cry of victory to embarrass the defeated party (see also in your face). Booyakasha was supposed to sound Jamaican or African. It means in context "Be quiet because I am right and you are wrong! I win! Feel embarrassed about it!"
"Her eyes were green, just like I said they were! Booyakasha!"