Something that is smudgy has marks or dirt on it that makes it not clean.
Smudgy things can make people feel like they need to clean them. It's not a good feeling.
Examples:
The window is smudgy and hard to see through.
She has a smudgy face after playing outside.
The paper is all smudgy from the marker.
I don't want to eat the smudgy cookie.
The whole room is smudgy and needs to be cleaned.
He has a smudgy handprint on his shirt.