Why does a pull down resistor eliminate floating input
Input pins often have a huge input impedance, in the range of mega-ohms. So only very miniscule current can and will flow into or out of them.

This means that only miniscule electrical noise is sufficient to change the voltage on the pin. This electrical noise can come from e.g. RF interference picked up from the environment. The pin and connected lines then act like antennas, and the very minute electrical power picked up by these antennas can be enough to affect the pin's state.

That's because whatever electrical noise gets picked up has almost nowhere to go, because the pin is so high impedance that it hardly does conduct away the noise at all.

Then, in every real circuitry, we have parasitic capacitances and resistances, both internal to the chip and external. This can cause minute charges to flow to the pin and also change it's voltage.

The function of a pull-up or pull-down resistor is to conduct away any undesired electrical charge so that it cannot affect the input pin's state.

For this, the resistance of the pull-up or pull-down must be significantly lower than the pin's input impedance, which isn't hard to achieve since the input impedance is quite large, see above. 

In more technical terms, we have a very high impedance source of noise and other parasitic signals, coupled to a very high impedance input. The pull-up/-down resistor basically gets connected in parallel to the input pin (and in series with the noise source) and reduces its effective input impedance which proportionally reduces the voltage the high impedance noise can induce.