What is the reason behind conditioning on an ancillary statistic ? Why?
Why do we want to condition on an ancillary statistic ?
Is it because by conditioning on an ancillary statistic, we can obtain something that is conditionally sufficient ??
I think Wikipedia explains it fairly well (read all of that section, including the batting average example). When one statistic (such as the batting average) is not sufficient, it may be sufficient in combination with (or conditioned on) an ancillary statistic (in this case, the number of at-bats). So you are basically correct that conditioning on an ancillary statistic (often) allows us to get something which is conditionally sufficient.
RA Fisher felt that the Fisher Information should also be measured conditionally on an ancillary statistic, so that it measure how much information beyond the ancillary statistic your statistic of choice gives you. So this could be another reason to condition on the ancillary statistic, though unfortunately it's often not practical.