Hello Gerry
"Como la gente grande" mean "As grown people" (adult people)
I would not say it means "great people" because the correct expretion would be "la gente grandiosa" (greatness, great people)
Example: A child tells his mom "Mamá! Quiero tener mis propias cosas, como la gente grande" (Mom! I want my own staff, just like grown people)
I hope this is usefull for you
Regards