From Wikipedia, "a multiplexer, also known as data selector, is a device that selects between several analog or digital inputs and forwards it to a single output line."
Each input take turns in connecting to the output line. There is only one input channel transmitting at a time. When an input is selected, it is as if there is a continuous path from that input to the output line.
So therefore, the line rate of the output channel must be the rate of which input it is connected to. For example, consider the figure above. If input B is selected, and its rate is 64 kbps, the output rate should also be 64 kbps.
But we know that in T-carrier system this is not the case. The rate of T1 is 1.544 Mbps because it is 24 channels of 64 kbps each plus 1 framing bit of 8000 samples per second.
Now that's the mathematics. But in terms of electronics, how could the multiplexer transmit at a rate higher than the input rate?