Expressions as Trees
You can think of any Wolfram Language expression as a tree. In the expression above, the top node in the tree consists of a Plus. From this node come two "branches", x^3 and (1+x)^2. From the x^3 node, there are then two branches, x and 3, which can be viewed as "leaves" of the tree.
The indices that label each part of an expression have a simple interpretation in terms of trees. Descending from the top node of the tree, each index specifies which branch to take in order to reach the part you want.