For example Witch Bolt says "Concentration, up to 1 minute" for it's duration. Is this real world or in game? How does the GM know when this ends other than the explicit conditions in the description?
The duration is in reference to in-game time and is measured by combat rounds (a full cycle of each combatant's turns). Each round of combat is 6 seconds, so in the case of Witch Bolt it would last 10 rounds of combat if concentration is not broken beforehand (6 second round X 10 rounds = 60 seconds/1 minute).
Located on page 181 in the PHB
In combat and other fast-paced situations, the game relies on rounds, a 6 second span of time described in chapter 9, "Combat."
All timings are in game
Player's Handbook never assumes real world timings. All timings like "1 minute", "1 hour", "8 hours", "3 days" etc. are supposed to be measured in game by DM. A DM is free to pause, slow or fast forward the game time. For instance, a typical "long rest" is 8 hours long, but in real world it is just a few seconds, like in a movie: heroes fell asleep - fade to black - new scene, heroes woke up.
When precision is important, DM can ask for initiative
A DM might ask players to "roll for initiative", switching the game mode, so to say. Initiative system assumes that all actions will be measured in "combat rounds", 6 seconds each. This allows the DM to track time more precisely. This mode is usually associated with combat and is described in PHB Chapter 9 "Combat", but DM might use it out of combat as well.
More information: How does time pass in combat?