Why/How do higher amp hour batteries for tools provide more power?
Batteries for power tools are made of a set of more-or-less standard primary "Li-Ion cells", typically of 18650 size. The cells came in different capacities, but overall the capacity is limited by current state of technology given the fixed cell volume. There are high-capacity cells, with higher energy density, and cells with less nameplate capacity.

One important thing for power tools is an ability to supply high peak currents to overcome start-up inrush currents for motors, and to handle load. So the battery cells must be of "high-discharge" type. High discharge means smaller internal resistance (ESR) and therefore require thicker foil electrodes and current collectors. As such, the overall power density (given the same volume of the cell) is smaller. That's why mass-produced power tools use cells of seemingly lower capacity, say 1500 mAh. 

Now, to increase total battery capacity they use a bunch of same cells (or series string of cells) in PARALLEL. This increases not only the overall capacity (6 cells of 1500 mAh give about 9000 mAh battery), but automatically it decreases the overall ESR (six ESRs in parallel result in 1/6 of the original ESR). Thus the battery becomes more "powerful" automatically (can deliver more current). So there is no artificial "current limiter" inside power tool battery pack (besides the fire protection), the limitation comes from property of individual primary cells used to make the battery.