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 feature Extra Attack which only activates on the character's turn. When someone else is the driving force of the action, the character is unable to capitalize on this aspect of his/her training. Even if they Ready the Attack action for use on someone else's turn, they only get one attack.
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. See the note on space below.
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 (when they have the most control over the battle on their turn)