Climbing shoes are small and much less bulky than everyday shoes to provide better precision, accuracy, sensitivity, and grip on climbing holds. In addition to this, the tightness and minimal material of the shoes, and the minimal dead space within the shoes, allow them to support the climbers and hold their weight when they need to stand on tiny foot holds.
Climbing shoes are created specifically to help maximize how climbers use their feet while climbing. For example, if a climber would need to do a heel hook on a small hold, in larger, everyday shoes, the climbers feet would just slip off the hold rather than actually hold their weight.
This is because normal everyday shoes have a lot more soft rubber on the bottom of the shoes, and a lot of dead space between the shoe and the foot. This soft rubber prevents climbers from being able to actually put their weight on small climbing holds, because the rubber is too soft to actually have any kind of grip on the holds, and it's too soft to hold the climbers weight, so the shoes just slip right off.
The dead space in everyday shoes is also an issue for climbing, it reduces the shoes effectiveness in certain moves such as heel hooking.
This is why climbing shoes are typically very tight and small, it just gives climbers the best support for their footwork.