See in SEP the entry on Intuitionistic Logic
The rejection of the Law of Excluded Middle does not means that intuitionsits adopt a three (or more) -valued logic.
According to Brouwer :
LEM was abstracted from finite situations, then extended without justification to statements about infinite collections. For example, if x, y range over the natural numbers 0, 1, 2, … and B(x) abbreviates the property (there is a y > x such that both y and y+2 are prime numbers), then we have no general method for deciding whether B(x) is true or false for arbitrary x, so ∀x(B(x) ∨ ¬B(x)) cannot be asserted in the present state of our knowledge. [...]
But to Brouwer the general LEM was equivalent to the a priori assumption that every mathematical problem has a solution — an assumption he rejected, anticipating Gödel's incompleteness theorem by a quarter of a century.
To see this, we need only reflect on the following Goldbach conjecture (GC):
Every even integer > 2 can be written as a sum of two primes,
which remains neither proved nor disproved despite the best efforts of many of the leading mathematicians since it was first raised in a letter from Goldbach to Euler in 1742.
Formalized intuitionistic logic is naturally motivated by the informal Brouwer-Heyting-Kolmogorov explication of intuitionistic truth.[...] From the B-H-K viewpoint, a sentence of the form (A ∨ B) asserts that either a proof of A, or a proof of B, has been constructed.
So, the rejection of LEM is motivated by the fact that we are not able to assert, for every mathematical problem A, that we posses a proof of A, or a proof of ¬A .