Rock climbing, generally, demands that the primary point of foot contact is beneath the big-toe area. The more movement that can be removed from the toes (as an appendage), the more stable the power transfer is, because you are now trying to support your body weight on a very small surface area. To do this, climbing shoes try to 'scrunch' your toes / foot - forcing direct power through the bones and deliberately limiting joint movement.
A secondary goal is to attempt to mould the shoe - like a glove - with the aim of limiting the shoe twisting around your foot (there's no use in having sticky rubber on the foothold, but your foot just spins inside the shoe).
The third objective is the continual compromise - compromising between providing a platform and soft sensitivity. You can't have both. A rigid platform is inherently easier to stand on/in, but removes the ability of the sole to mould around a feature/grip as well as removing sensitivity expressly because it is taking a lot of the workload.
All these things contribute to better performance.
When you start out climbing, your toes are unaccustomed to the confines of scrunched toe-boxes, AND your foot muscles are weak, your foot having spent it's entire life flat on the ground, and is, thus, not yet strong enough across the arch and ball, to cope with the small surface area now being expected to support your weight. As you climb more, and your feet get progressively stronger, your footwork will want to feel more of the rock as well as have more flexible options, literally. If the foot-grip is sloping or curved, the shoe needs to bend and mold to that shape, and to do this needs to be less rigid. But rigidity requires less strength, so sacrificing rigidity needs to happen alongside stronger foot muscles. Conveniently, the types of foot grips most commonly found on the types of routes and problems in your early climbing days will tend to be bigger, flatter and more angular. As you progress through the grades, the foot grips will become less pronounced and require more sensitive shoes. Style also plays a part. Vertical routes tend to require flatter, stiffer shoes while the more the route overhangs the more down-turned of a shoe is beneficial. Similar rigid trade-offs are found here too - the weaker you are, the more a stiff shoe will help you. It is important to understand that this is ALL relative to the individual. There is no such thing as "easy routes" - only easy routes in comparison to something harder FOR YOU! IE a 9A climber can still be "weak" *and require a specific shoe to aid them)...it's just relative to what is considered "hard" to them / you.