When something sists, it means it is in a place and doesn't move from there.
If something sists, it is usually staying in one place for a while.
Examples:
The cat sists on the mat.
The book sists on the shelf.
The sun sists in the sky during the day.
The snowman sists in the yard all winter.
The idea sists in my head and won't go away.
The secret sists in the box and no one knows what it is.