Loosely speaking, constructivism means seeing is believing. The principle is that only those ideas and properties exist for which we can exhibit an appropriate configuration in terms of an agreed way of representation. For example, if it's agreed to represent real numbers via Cauchy sequences of rational numbers, then the square root of 2 exists only after one has constructed an appropriate Cauchy sequence. Before that, the square root of 2 does not exist, however esoteric this may sound.
The allowed ways of construction differ in different flavors of constructivism. In most cases though, the set of natural numbers is either assumed to exist or allowed to be constructed, either as actual (completed) infinity or as potential (incomplete) infinity. Moreover, instead of carrying out a construction (e.g. that of a square root), it may be agreed that it suffices just to provide a feasible method for the same.
Truth values have to be constructed too. Here the "appropriate configuration" is the concrete proof, and the "agreed way of representation" is the allowed forms of proof. A statement does not have a truth value until it has either been proved or disproved (or until a feasible method has been provided that would certainly result in a proof of truth or falsity).(6) If there is no such truth value to observe, it simply does not exist for a constructivist, leaving the statement undecided.(7)
To prove "A or B", we need to prove at least one of them. This requirement lies at the very heart of constructivism, and follows from a "seeing is believing" interpretation of logical disjunction. Accordingly, to be able to say that "A or (not A)" is true, we need to either prove A or disprove A. This is different from classical mathematics where "A or (not A)" is always true in itself, since there it is taken for granted that every statement has a truth value, even if nobody can actually observe it. Similar applies to "not (not A) ⇒ A": in constructivism, the left hand side only means we've demonstrated that there is no way of disproving A; but it does not necessarily imply that A is true, or that A has a truth value at all.
Adhering to the principle of constructivism lends constructive mathematics certainty and confidence, and leaves little room for unpleasant surprises like paradoxes or contradictions. Eventually, it's hard to imagine a more obvious and tangible evidence of existence than that of a constructed representation. The price we pay is that proofs tend to become unusually cumbersome.(8)
— ◊ —
(6) What does "disproved" mean in constructivism?
A statement is disproved after a contradiction has been derived from the assumption that it's true (= proved). This is how the truth value "false" is defined. The following are synonymous: "A is false", "A is disproved", "not A". To paraphrase, a statement is false if and only if it has been proved that the situation described by it cannot occur.
(7) Assuming that the square root of 2 has not yet been constructed, can we say that it does not exist, or only that its existence is undecided?
If existence is meant literally, then given the assumption, the statement "the square root of 2 exists" is false. However, if by "exists" we actually meant "can be constructed", then the statement is (currently) undecided.
(8) Why are constructive proofs cumbersome?
I believe it has to do with our education, namely that we have been trained in classical mathematics from an age of 6 or so. As a result, textbooks on classical mathematics can be written in an informal style where many details that would clutter the main line of thought are omitted. It does not compromise rigor, since it capitalizes on the readers' solid understanding of and intuition about the fundamentals. Alternative mathematical theories don't have this luxury. In their textbooks, theorems and proofs are stated either in great detail at the expense of increased clutter, or with less details risking that the audience gets confused or misunderstands what is written. Either way, the exposition is likely to be difficult to follow.