Why does armature current increase when load on the DC motor increase?
From Wikipedia's Counter-electromotive force.


  The term back electromotive force is also commonly used to refer to the voltage that occurs in electric motors where there is relative motion between the armature and the magnetic field produced by the motor's field coils, thus also acting as a generator while running as a motor.


Simple version: When a motor is accelerated from zero speed it also acts as a generator. 


  This voltage is in series with and opposes the original applied voltage and is called "back-electromotive force" (by Lenz's law). 


The generator voltage opposes the driving voltage.


  With a lower overall voltage across motor's internal resistance as the motor turns faster, the current flowing into the motor decreases.


The faster the motor spins the less current it draws.

Back to your question:


  Why does armature current increase when load on the DC motor increase?


When the load increases the motor slows down. This reduces the counter EMF. This allows a higher current to flow.


  Does this load somehow decrease the resistance of the armature causing it to draw more current? Is it due to the torque the load will give to the motor?


No, but it looks like it.







simulate this circuit &ndash; Schematic created using CircuitLab


In Figure 1 (a) the motor is stalled so there is 1 Ω across the 12 V supply and 12 A will flow.
In (b) the motor is running close to half-speed and the back EMF is 5 V. This means that there is 12 - 5 = 7 V available to drive through the motor's 1 Ω resistance so current falls to 7 A.
In (c) the motor is running close to full speed and the back EMF is 10 V. That means there is only 2 V available and current falls to 2 A.