Turing machine program states are conventionally represented with letters: A, B, C, etc. The starting state is A.

Now suppose you are running a Turing machine program from the beginning. The only state it has visited so far is state A. It runs until it reaches a state that has not been visited yet. What state is it? B? C? D? According to "tree normal form", the name for that next state just is the earliest unused state name, which in this case is B.

Visited states so far are A and B. Run until an unvisited state is reached again. What is its name? C? D? E? Tree normal form says that the state will be called C. And the next newly visited state will be D, etc. In general, the canonical form for a Turing machine program will be the one that puts initial state visits in alphabetical order. (This concept also applies to the multi-color case.)

It's not possible to tell of an arbitrary TM program whether or not that program is in tree normal form. But this proof doesn't look at arbitrary programs. It generates candidate programs by tracing out the normal tree starting from the root, thereby bypassing non-normal programs altogether.

That is what "essentially different" means here.