What does the output current rating from an H-Bridge's datasheet really mean?
Recently, I bought the H-Bridge VNH7070BAS from ST because it would
  drive up to 15A


It will not drive anything like this level of current for anything more than a few micro seconds before the automatic current limit circuit operated.

The internal transistors together contribute circa 0.1 ohm impedance in the current path and, at (say) 10 amps continuous, the internal power dissipation is 10 watts and an SO-16 package can no-way handle this amount of internal power dissipation.

At best, the thermal properties imply around 30 degC rise per watt dissipated and, at 10 watts, that's a temperature rise of 300 degC and totally unfeasible for the device even on a large heatsink.


  when I test it, it only reach 6A and it shut down


That's an internal power of around 3.6 watts producing a temperature rise of around 108 degC in an ambient of maybe 25 degC that rapidly rises locally to the device to maybe 50 degC and hey presto, the silicon junction soon rises to above 150 degC and the device shuts down. Thermal problems.