BJT gain.Width of base region vs collector current
At first, you have to realize that the emitter current - and accordingly the collector current - is controlled by the base-emitter voltage Vbe (see Shockleys famous exponential equation). I am aware that in some textbooks the BJT is described as a current-controlled device, but that is a simplified description which cannot explain all observable effects. The base current does exist (unfortunately) and it is taken into account for calculating the base bias circuitry - but it is kind of by product.
(Barrie Gilbert: ...the base current is purely incidental - it is best viewed as a „defect“).

Now - what happens if the voltage Vce across the BJT increases? This increase results in a broader depletion layer between collector and base (Vcb increases). As a consequence, the space charge region between base and emitter (forward biased by Vbe) decreases. Therefore, the electrical field strength within this area is increased (constant base-emitter voltage) - and more charged carriers are able to move to the collector. This effect somewhat increases the collector current (and reduces the base current for a constant emitter current because of Vbe=const.). 

Note: This happens even in the case that the base current is kept constant (Ib=const.). Hence, this is one of some proofs that it is the voltage Vbe which controls Ic.