Each analog multiplexer connects one of 16 pins to an Arduino analog input. In the diagram each of those 16 pins has a pull-up resistor. Those are only necessary if your "sensor" is a variable resistor: LDR for light, Thermistor for temperature, PDR for pressure... In those cases the other end of the sensor goes to Ground and the sensor/resistor combination acts as a voltage divider.
If your 'sensor' is a potentiometer you should leave out the pull-up resistor. Connect the ends of the pot. to +5 and Ground. Connect the wiper of the pot. to the input pin.
If your sensor is some other form of voltage source (0 to 5 volts) you should also leave out the pull-up resistor.