How does a HCF4051BE multiplexer work, at a very simple level?
If you want to know what it does for you (rather than how it works internally) you can visualize the chip as an eight-position rotary switch, COM OUT/In is the common contact, CHANNEL I/O the 8 individual contacts.

The position of the switch is determined by the digital value on the pins A-B-C. The INH input (when active) sets the switch to a 9th position that is not externally connected.

In practice the switch is not ideal: it has a non-connectible (and somehwat non-linear) series resistance, it can not handle voltages outside the Vee - Vdd range, etc.