Constructivism is a view that has its origins in the philosophy of mathematics. Roughly, it's the idea that there are no "external" mathematical entities that provide truth conditions, or correctness conditions, for mathematical propositions. (So it contrasts with standard approaches in physics and engineering, which treat the external world as providing truth/correctness-conditions for scientific models and theories.)
Adherents of constructivism in its stronger form reject the validity of certain types of indirect proof, on the grounds that unless you have a direct proof that "constructs" the relevant proposition, it's invalid to infer to it's existence - because that would be positing a type of externality to mathematical entities that constructivism denies. The most famous strong constructivist of this sort was Wittgenstein.
More generally (ie beyond phil of maths), constructivism is (roughly) the idea that certain procedures or processes are validating of their outcomes without there being any independent correctness conditions for those outcomes. What makes this more than just a defence of the arbitrary is that the "certain procedures or processes" have to be specified in a manner that makes it plausible to regard them as sufficient to validate their outcomes. In constructivist accounts of scientific method, it is the account of how scientific method renders observations and experience tractable, and amenable to theorising in a way that has problem-solving power, that explains why the outcomes of scientific investigation should be treated as knowledge rather than mere arbitrary conjecture. But this idea is not obviously applicable to RPGing.
But there is a simple application of the constructivist idea - that is, that certain procedures validate their outcomes without the need for independent correctness conditions - that does have relevance for RPGing. This is the example of a fair bet: if the bet is fair, and participation truly voluntary (and if we ignore bigger contextual questions like where did the participants get their money from to make the bet in the first place), then (i) there is no correctness condition, independent of the process, that tells us who should own the stakes after the bet is resolved, and (ii) once the bet is resolved, the resulting distribution of the ownership of the stakes is validated *simply by the fact that it is the outcome of a fair bet.
The example of betting generalises to a lot of game play, including RPGs. One frequent use that RPGs make of dice rolls is to produce outcomes - changes to the shared fiction - that are accepted as valid by everyone because they follow from the roll and with no independent correctness condition. The best known example is combat: thus, even if the PCs eke out an unlikely win against the giants, we don't say normally say "Hang on, that makes no sense - we'd better ignore the dice rolls and substitute our independent judgement as to how this should have worked out." Rather, we treat the dice rolls as sufficient to validate the outcome.
The extension of this sort process-based/constructivist approach beyond combat is a recurring issue of disagreement. But there are a number of RPGs that make it pretty crucial across the board - some mainstream ones like 4e D&D, and some "indie" ones like Apocalypse World and many of its offshoots, Burning Wheel and its associated family of games, etc.
The idea of process-validated outcomes can also extend beyond dice rolls. For instance, in standard D&D play if a player correctly follows the rules for writing down a memorised spell on their PC sheet; and then correctly follows the action declaration rules for making it true that, in the fiction, their PC casts the spell, then the spell takes effect in accordance with its rules. The same thing happens in Prince Valiant when a player spends a "storyteller certificate" that they have earned in accordance with the rules - for instance, when Sir Morgath's player spent a certificate to Kill a Foe in Combat then it meant that he bested Sir Lionheart (the greatest knight in Britain) and then when he did it again he bested the giant crocodile while swimming in his armour in the Black Sea. The fact that the procedure validates the outcome means it's not open at the table for anyone to say "Hang on, Sir Morgath only has so many combat dice, whereas Sir Lionheart and the crocodile have many more, so he couldn't have beaten them!" Rather, just as in the case of combat resolution via dice rolls, the fiction needs to accommodate the outcome. (In the case of Sir Lionheart, for instance, Sir Morgath's player narrated it thus: As my lance splinters on his shield, a small bit of wood flies through his visor into this eye and brain, killing him.)
One recurrent source of disagreement among RPGers is the extent to which the GM is entitled to suspend procedures, or to ignore their outcomes, in order to make sure that the correct outcome is arrived at. This includes debates about "fudging", but goes beyond it: for instance, can a GM veto a player's attempted spend of a stortyteller certificate, or a player's declaration that their PC casts a spell? Those who are inclined to say yes - who see the procedures as sometimes, even often, useful heuristics but don't regard the outcomes as being validated by the procedures alone - are not constructivists (at least in the domain of RPGing).
(Internal realism is a view in philosophy of science that was advocated for a while by Hilary Putnam. It's not far wrong to regard it as a version of constructivism: though it is a bit less concerned with process than constructivism in the strictest sense, it has a similarly close connections to Kantian ideas. I don't think it helps us much in understanding RPGing.)
