The Bugbears' Long-Limbed trait says:

When you make a melee attack on your turn, your reach for it is 5 feet
greater than normal.

This means that when it is not a Bugbear's turn, his reach is five feet shorter than it is when it is his turn and he's making a melee attack. Does this mean that the bugbears' arms extend out from their bodies like Reed Richards on their turns when they strike out? Could it happen outside of combat when turns aren't being tracked? If not, maybe it's a response to a threatening situation?

Why I care: I want to know if my Bugbear has stretchy arms because it's important to the narrative if such an odd physical feature exists. My GM will want to describe the action in detail.

Answer parameters: Any relevant RAW, Errata or Sage Advice comments either confirming this or explaining why the arm stretchiness isn't a thing, but I'll take designer opinions, too in the absence of official rulings or RAW. If there's any reference in prior editions of D&D to bugbears having arms that physically get longer on their turns than they are when their turns are over, then that would be nice to know, too.

They probably don't stretch

The name of the feature: "long-limbed" seems to merely be referring to the length of the arms themselves. If stretchiness was an aspect of bugbear physiology, that is something that would probably be mentioned in the description of bugbears either in the Monster Manual or Volo's Guide to Monsters.

Why only on the bugbear's turn then?

To start with, turns are an abstraction for the sake of ease of play, but narratively they can be understood as the aspect of a round where the character has the most control over the fight. Remember that everything that happens in a 6-second round has no particular order in-universe. So all this means is that the bugbear can only leverage their long limbs to attack a creature who is further away during their opportunity to influence that 6-second window.

You can see a similar phenomena with the rogue feature Sneak Attack which only activates on the rogue's turn. When someone else is the driving force of the action, the rogue is unable to capitalize on this aspect of his/her training.

Out of combat

When not in combat (or using turns for some other event), the reach of a creature rarely matters. Instead, arm length itself is what's important (for reaching things). Arm length is something not touched on at all in the game, so it will be up to a conversation with your GM how long a bugbear's long limbs are.

However, it is worth noting that just because a bugbear can attack something twice as far away, does not necessarily mean their arms are twice as long as other medium humanoids. From the Basic Rules:

A creature’s space is the area in feet that it effectively controls in combat, not an expression of its physical dimensions.

At any given point in the 6 seconds that make up a round, a creature need not occupy a specific location in the space it controls. Their position ebbs and flows as the battle rages on. It is easily possible that a bugbear's long arms allow it to continue to control that medium sized space without even having to be within that space entirely. With this explanation, the bugbear might just be temporarily ebbing into another space to make the attack and flowing back.

It's up to you as a player/DM

There is no description of how Long-Limbed works in-universe. It just does. As a player or DM, you are entirely free to describe what this looks like at your table as long as you use the existing mechanic of how it works.

That's the joy of roleplay!