Trees planted in fall get to enjoy the milder weather with none of the pressures of growing up too fast. They can focus their energy on growing more roots, rather than growing new foliage.
The soil is actually warmer then than in spring. That’s because it’s been warmed up all summer long and stays warm—long enough for a tree to establish roots and absorb nutrients before it goes dormant.
Roots start growing as soon as they touch moist soil and continue growing as long as the soil temperature stays above 45°F (a baseline that can extend into winter with a good layer of mulch).
While all new trees (even drought-tolerant trees) need plenty of water to become established, the task of watering is much less onerous in fall since it usually comes with more wet weather.
Newly planted trees need consistent watering right up until the ground freezes. That means you still need to water them if you end up with an unusually dry fall, but not as often (or as much) as you would if you’d planted in spring (going into the heat of summer).