Exact meaning of the Gandalf quote, "He that breaks a thing to find out what it is has left the path of wisdom."?

What is the exact meaning of following quote (it belongs to Gandalf the Grey):

He that breaks a thing to find out what it is has left the path of wisdom.

I have a problem especially with understanding what it is has left.

Sounds like you are not seeing the way the sentence breaks into phrases. Think about it like this:

He that breaks a thing / to find out what it is / has left the path of wisdom.

Loosely, it means: "If you break something in order to fully understand it, you are a fool."

It means:

If you break something to find out what it is, then you have left the path of wisdom.

You are misparsing the actual constituents here: “is has” is not part of the same constituent, but rather two separate pieces of two completely separate constituents. That is is actually part of the noun phrase serving as the sentence subject, while that has is actually part of the verb phrase serving as the sentence predicate.

Simplified, a sentence is this:

sentence = subject + predicate

So the is falls at the end of the subject while the has falls at the start of the predicate.

If you feed Gandalf’s sentence to the Link Parser and examine the resulting constituent tree it generates, all should become crystal clear because the inherent structure of the sentence is revealed:

He that breaks a thing to find out what it is has left the path of wisdom.

Constituent tree:

(S (NP (NP He)
       (SBAR (WHNP that)
             (S (VP breaks
                    (NP a thing)
                    (S (VP to
                           (VP find
                               (PRT out)
                               (NP (SBAR (WHNP what)
                                         (S (NP it)
                                            (VP is)))))))))))
   (VP has
       (VP left
           (NP (NP the path)
               (PP of
                   (NP wisdom)))))
   .)

Do you see now why the is and the has are each part of completely different logical constituents?

There are two top-level constituents here: a noun phrase (NP) acting as the subject of the sentence (S) and a verb phrase (VP) acting as its predicate. The subject begins with he while the predicate begins with has.

You can find the meaning of the rest of the ALLCAPS constituent tags here at the Penn Treebank Constituents list. It defines the phrase-level tags used in this tree, in order of occurrence left to right, as:

• S — simple declarative clause, i.e. one that is not introduced by a (possible empty) subordinating conjunction or a wh-word and that does not exhibit subject–verb inversion.
• NP — Noun Phrase.
• SBAR — Clause introduced by a (possibly empty) subordinating conjunction.
• WHNP — Wh-noun Phrase. Introduces a clause with an NP gap. May be null (containing the 0 complementizer) or lexical, containing some wh-word, e.g. who, which book, whose daughter, none of which, or how many leopards.
• VP — Verb Phrase.
• PRT — Particle. Category for words that should be tagged RP.
• PP — Prepositional Phrase.

Just my two cents. I used this sentence to see how link-grammar library copes with it (I'm not associated with the library / it isn't a commercial product) and here are the results:

He that breaks a thing to find out what it is has left the path of wisdom.
Found 360 linkages (86 had no P.P. violations)
    Linkage 1, cost vector = (UNUSED=0 DIS= 7.00 LEN=34)

    +-----------------------------------------------------Xp----------------------------------------------------+ 
    |      +------------------------------Ss------------------------------+                                     | 
    |      |              +-------MVi------+                              |                                     | 
    |      +------Bs------+----Os---+      |     +----Osn----+--Bsdt-+    |       +-----Os----+                 | 
    +--Wd--+--R--+---RS---+    +Ds**+      +--I--+--K--+     +-Rn+-Ss+    +---PP--+     +Ds**c+-Mp-+--Ju--+     | 
    |      |     |        |    |    |      |     |     |     |   |   |    |       |     |     |    |      |     | 
LEFT-WALL he that.j-r breaks.v a thing.n to.r find.v out.r what it is.v has.v left.v-d the path.n of wisdom.n-u . 

As you can see, it links he (the subject) to has left via Ss link, which, simplifying it, gives he has left the path of wisdom, just as other answers had already suggested.