This is a topic of whole chapters in botany textbooks. I tried really hard to find some simple diagrams/pictures to help do it in a few words.
Basically the roots of a tree are spread out all around the tree just like the branches are above ground. Each root has many divisions (just like the branches on moe so and much tinier at the ends) and it is near the very ends that water and nutrients are "sucked" up for lack of a better word and passed along up through the trunk in a series of "tubes" (like straws) that then carry the moisture and nutrients out to each branch and leaf.
If a root grows around other roots it can choke the other roots and damage itself causing less and less flow up the "tubes" to that part of the tree. This might cause part of the tree to begin to die.
Diseases and pests can often attack the roots and prevent the water and nutrient flow as well with the same partial death or damage to the tree.
Sometimes there is physical damage to the roots from critters like voles or moles that damage the roots or damage the bark at the point where the tree emerges from the ground which stops the water and nutrients from passing up the tree.
Rarely a lightening strike to the tree will damage both the central core of "straws" on one side of a tree and the roots where the lightening discharges into the ground and leave half the tree damaged but the other half growing.
Some help in visualizing the tree roots and tree here:
transpiration and here:
Lesson 6 | Tree Kit | Illinois Natural History Suvey