Actually, that isn't quite true. With an electric motor power stays about the same throughout the range of rpm, but as the number of rpm increases, torque decreases. The same is true the other way around, as the number of rpm decreases, torque increases. It's actually impossible to keep torque and power constant while varying the rpm.
Electric motors usually can get away with no transmission, saving weight and mechanical losses due to friction. It might also be less expensive to produce just a larger motor than a smaller one and a transmission. That said including a transmission can certainly make better use of the available power in many situations. Applications like ours would benefit significantly from this, providing better acceleration and a higher top speed.
One example of an electric motor paired with a multi-speed transmission can be seen on the Brammo Empulse electric motorcycle.