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The increase of current by adding a second branch of circuit is based on the condition that the supplied voltage remains unchanged as before.
A second branch to a voltage source (e.g. a battery) can be treated as another independent circuit connected to the voltage source. In that sense, the voltage source now needs to deliver current separely to two independent circuits. It is clear that the total current given out by the voltage source is the sum of the current required by each circuit.
This situation can also be explained by Ohm's Law, the addition of a new branch forms a parallel connection. The total resistance of parallel connections must be smaller than the resistance of individual branch. By Ohm's Law, lowering of the resistance will increase the current flow for a given voltage supply.

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