The claws of a tiger enable it to get a firm grip on the back of its fleeing prey. If it has smooth paws like a human's hands its prey would easily avoid being captured. Its teeth are sharp in order that it can deliver the fatal bite to its prey and to enable it to bite through its prey's integument and to rip of muscles. All predators have weapons that enable them to catch, kill and eat their prey. Prey equals food and food provides nutrition.
Any tiger born without the genes that code for both of these traits will not be able to hunt, or even eat, and they will starve to death before they can reproduce. Therefore only tigers that have both traits are able to survive and pass on their genes. The elimination of individuals without either of these 2 traits is natural selection. It is called negative selection. Tigers did not evolve sharp teeth and claws. They inherited both features from their ancestors. Their ancestors evolved from a tree climbing, meat eating predator and they needed sharp teeth and claws to hunt as well.
Claws in fact evolved evolved through mutation and natural selection in reptiles, which were the ancestor of mammals. Claws are found on reptiles, birds and mammals. The reptile that first evolved claws had a mutation that give them claws. Once claws evolved, they were passed along to future generations because they allow individuals with claws to outcompete those that do not have claws. Their descendants outnumber those without claws, so that eventually those reptiles that did not have claws eventually died out. The ancestor of tigers 65 million years ago was a shrew-like mammal that spent most of its time underground, and it inherited claws that allow it to dig burrows from its ancestor. Since mammals have been living inside burrows since they first evolved about 220 million years ago, sharp claws that can be used to dig have existed for at least that long.
In contrast, sharp teeth evolved even earlier, in the fish ancestor of the tigers. Therefore tigers are not unique among mammals in having sharp teeth and claws. In fact mammals without sharp teeth or claws (e.g. humans) have only recently evolved unsharp teeth and claws (or nails) relatively recently, Our closest relative, such as monkeys and apes, still have sharp canines teeth. We are primates and primates evolved nails quite a bit earlier than that.
Defense and feeding.