Inserting into a BST is searching for a spot that doesn't exist yet. Compare the new value at each node, smaller goes left, bigger goes right, until you hit an empty position. That's the new node's home.
Because the walk follows the same ordering rule as search, the promise (smaller-left, bigger-right) survives automatically. On a balanced tree the walk is O(log n), the same halving that makes lookup fast makes insertion fast too.