Planes tilt while turning to maintain airspeed, altitude, and provide the best passenger comfort.
If you have seen any acrobatic flying, you already know that it is possible for aircraft to perform amazing feats. They can fly upside down, spin, stall in midair, dive straight down, or accelerate straight up. If you are on a passenger aircraft, you are unlikely to experience any of these maneuvers.
Just one pilot has done a successful barrel roll with a Boeing 707 during a test flight. You can see test pilot Tex Johnson's description and some old video: here.
Extreme maneuvers like this stress the aircraft, are risky, and above all are not pleasing to passengers. An easy banking turn saves fuel and is much safer for the equipment and the passengers.
The centripetal acceleration the passengers feel pushes them toward the floor and not side to side (or up!).
By tilting the plane, you use the lift force to push it in the desired direction.
Airplanes are not like cars, that rely on friction to redirect the momentum and make a turn. Instead they are a bit more like racing motorbikes.
Bikers tilt their bikes to do high speed turns, using the normal force of the road on the bike to create a torque in combination with the weight of the bike (when tilting the bike, normal and weight are no longer in the same line).
Airplanes do it similar but different. The shape of an airplane creates a net upwards force by forming turbulence and vortices in the air around itself. This lift force is perpendicular to the floor of the airplane. When the airplane tilts, this lift force is now pointing diagonal, part up and part towards the direction you are going to turn.
At this point, see how now the airplane has less lift in the upward direction: to keep altitude, while doing a turn, the plane must accelerate to compensate.