Bent climbing shoes are called aggressive climbing shoes. They are bent, and many times also have a curve in the direction of your big toe, to give you more power when standing with your toes on small edges, and to give you more pulling power with your toes. Imagine a piece of paper on the floor in front of you, put the tips of your toes on it and pull it towards you - aggressive climbing shoes make it easier to pull your body weight like this on climbing holds, they allow you to pull yourself closer to the wall and keep you there.
The bend helps you engage more and put more force onto the tip of your shoes, allowing you to add more tension and be more stable on tiny edges that otherwise could be very difficult to stand on.
Aggressive climbing shoes are also great for overhanging climbing, since the added tension and force you can exert with your toes can help you keep your feet on the wall, which prevents you from cutting loose and hanging from your hands, which can be very tiring and costly and could cost you finishing the route.
Climbing shoes are bent downwards to enable you to put more weight on very small edges. By being bent downwards, you are able to apply more force on the small edge without the shoe slipping off.
Bent climbing shoes are normally better for bouldering and overhanging routes, as they allow you to have a better grip on the holds, which enables you to pull harder on your toes and keeping your body close to the wall. All of this allows you to reserve more energy in your arms.
Though they are mostly used for bouldering and overhanging routes, they are not limited to them. There are many climbers who prefer aggressive downturned climbing shoes over neutral ones for sport climbing also.
Another element of aggressive climbing shoes is a slight twist in the direction of the big toe. This allows you to put even more force and power onto the toes, allowing you to have an even better stance on those very small edges.
