How do brushless motors provide more torque and longer battery life to cordless drills?
A drill manufacturer claiming higher torque and efficiency from their brushless motor is probably telling the truth - however that doesn't mean being brushless is the only reason for the improvement. They could put a really bad brushless motor in there that's no better than a high quality brushed motor, but why would they?

All else being equal, a brushless motor should have more torque because none is wasted in friction between the brushes and commutator, and higher efficiency because no power is lost in brush heating and contact arcing. Without having to worry about burning up brushes, a brushless motor can handle more current - and since current creates torque that means a brushless motor can produce higher torque without having its lifespan reduced.

But that's not all. The weight and space saved by not having brushes and a commutator leaves more for copper windings, stator iron and rotor magnets - the parts that actually create torque. Thus a brushless motor can be more powerful than a brushed motor of the same size. 

A brushless motor can also spin faster because it doesn't suffer from brush bounce, has no commutator to fly apart and no armature windings to get flung off. The rotor can be made from a single cylinder of Neodym magnet, or several magnets bonded to a core and wrapped in Kevlar. Whereas brushed motors typically max out at around 20,000rpm, a well made brushless motor might manage 50,000rpm. This higher speed can then be geared down to produce more torque.

Further improvements are mostly due to the use of better materials - eg. Neodym magnets instead of Ferrite, thinner laminations made from higher quality silicon steel. The same improvements could be applied to a brushed motor, but it would still be limited by the brushes. Adding higher quality brushes would probably make it cost more than an equivalent brushless, so why bother? 

Finally, a brushless controller can optimize its timing for best efficiency, speed or torque - whereas brushed motors usually have fixed timing which is optimized for one speed and one direction only.