Why do fingerprints look like this and why do people need them?

A person does not need fingerprints. The fact that a person has them is just a confluence of factors.
Here it should be clarified that fingerprints are a multifactorial thing, and each of the factors is needed for an independent reason.
Papillary pattern on the fingers - increases the roughness of the skin, increasing tenacity and sensitivity.
Sebum, which leaves a pattern on almost everywhere, is a natural moisturizer. Without it, the skin dries up and loses its elasticity. By the way, it is very funny to look at modern cosmetology. Half of the products in it are to remove the natural moisturizing agent, and the other half is the new moisturizing agent. Great business.

1) why they look like this: about the sixth week of embryonic development, swelling and bubbles appear on the fingertips and palms of the embryo. It is believed that the larger the bubble, the "more complex" the pattern will be. The simplest is an arc, then loops, then curls (there are more of them, but these three are the main ones). By the way, they are also on the feet.
2) First of all, they were needed by our ancestors living in trees - to improve tactile sensitivity, which is important in the way they move in space. For humans, palms (feet too, but we limited their sensitivity to shoes, asphalt, etc.) serve as the main organ of touch - due to these folds, the surface area and the number of mechanical receptors are increased. If you look at the projection of body parts onto the sensitive area of ​​the cerebral cortex, you will see that the same amount is occupied by the face and lips, the rest of the body has much less. Well, such a strong connection with the cortex determines the importance of developing the brain through touch - this is already a developmental psychology.