why reduced resistance and increased current result in an increased amount of heat
All of this relates to two things: 

Ohm's Law:  \$ R = \frac{V}{I}\$

Joule Heating \$ P_\text{Heat} = V \cdot I\$

The first tells us that if we keep the voltage \$V\$ a constant, the current will increase when the resistance decreases. This makes sense since the resistance is a measure of how hard it is to have current flow from one node to another - if it resist less, more current can flow.

The second then tells us that the power increase with current and with voltage. If we keep the voltage constant, but increase the current, the power will increase. 

In a resistor, all this energy is turned into heat. Thus, if we have more power being dissipated in the resistor, it will get warmer if it has a lower resistance (given it has a constant voltage across its terminals).