To make something more advanced or complex, or to make someone seem more advanced or complex.
The word sophisticate can make people think of fancy things, or of people who know a lot about fancy things.
Examples:
She likes to sophisticate her desserts with fancy decorations.
He tries to sophisticate his image by wearing fancy clothes.
The chef will sophisticate the simple dish with exotic spices.
I don't think I can sophisticate my toy car.
He tried to sophisticate his treehouse by adding a swimming pool and a library.
The magician will sophisticate his act with a new trick.