What is the meaning of Sentence and how it constructs

Although he had never taken piano lessons, anytime that Jeff, who was unusually intelligent, saw a piano, he was able to play a song.

If you want to identify the 'core' of the sentence, you must first remove all the optional dependents of the 'main' verb; in other words remove all the adjuncts, which in this case are:

[1] "although he had never taken piano lessons" (adjunct of concession)

[2] "anytime that Jeff, who was unusually intelligent, saw a piano" (adjunct of time)

That leaves the 'core' as:

he was able to play a song

The sentence has three-level depth:

The outermost sentence(1) is

Although he had never taken piano lessons, he was able to play a song.

The middle depth clause(2) is:

anytime that jeff saw a piano

The inner clause(3) is:

who was usually intelligent

The sentence structure is that (1) contains (2) and (2) contains (3).

Understand from the outer side to the inside by the level. First look at only (1). After you understand (1), add(2) into (1). And then, go to the (3).

The (3) is to describe the quality of jeff which is mentioned in (2).