You're correct in your understanding of the lyrics...it's like "he created this feeling in me" or "he made me feel like dancing and crying" The verb "got" is very versatile in English and we use it for so many meanings! Great question!

"Get" has hundreds of meanings but all are related to "acquisition". So, "I got sick" means I acquired some sickness. "He got me crying" means he did something that caused me to acquire crying behavior. "He got me feeling happy" means he behaved in a way, or said something, or did something that caused me to acquire a feeling of happiness. The verb "get" is so flexible that it is almost possible to speak English knowing no other verb. It's not a good idea to do that though. There is almost always a better verb to use.