If you look at something searchingly, you look at it very carefully, like you are trying to find something.
When you look at something searchingly, it can make people feel like you are very interested or curious about it.
Examples:
She looked at the map searchingly.
The detective examined the clues searchingly.
He searched the room searchingly for his lost toy.
The cat stared searchingly at the bird.
The scientist studied the rock searchingly for signs of life.
The teacher looked searchingly at the students to see who was paying attention.