How do batteries connected in parallel give more current than batteries connected in series?
As others note "can" and "will" usually differ.   

Imagine each battery had a chemical to electrical conversion capability such that it COULD deliver up to 0.5A.
If you connected a 1 Ohm load, Ohm's law would allow 1A IF the battery was able to supply it. But, as the battery was only able to supply 0.5 A max you'd see
V = IR = 0.5 x 1 = 0.5 V across the resistor.
ie the battery voltage would sag due to its limitations. 

Now use 3 similar capability batteries  in parallel.
Ability is now "up to 1.5 A" and
actual with 1 Ohm load will be 1A, as expected. 

In situations like this which seem to not make sense, work through the problem and see how the resultant "needs" match the capabilities.
Here, with one cell, if it CAN provide 1A or more then all is well.
If the per cell capability is smaller than the need, work out what is impossible.
Here if cell can make 0.5A then.

V = IR = 0.5 max x 1 Ohm = 0.5V is false (if battery is at 1V)
R = V/I = 1/.5 = 2 Ohms = false 
I = V/R = 1/1 = 1 A = false if max = 0.5A.
Choose any 3 :-) - all false.
Now use 3 cells and try again.
Ah!