Pitches add up.
Depending on other factors, you can greatly reduce the likelihood of a future arm injury if you throw more than 110 pitches in a game.
Each pitch a pitcher throws puts an enormous amount of stress on their body and thus there are limits to the amount that a human can throw on a given day at max effort without increasing their risk of injury substantially.
When a team like the Mets determines the number of pitches a pitcher can throw in a game, they are looking at a number of factors in consultation with trainers and medical professionals. They are looking at medical history, body type, and the players history. If a player is slight or has a lengthy injury history, they will be much more conservative with the number of pitches they allow him to throw in a game because throwing more puts you at greater injury risk.
As far as what it takes to add additional pitches to your endurance, getting into better shape might help, but most likely going through seasons injury free may allow you to increase your future load.
There are numerous other factors that may change the number of pitches a pitcher is allowed to throw as well:
Types of situations a pitcher finds themselves in during the game. If a pitcher is "cruising" and does not throw a lot of pitches with base runners on base, they may be allowed more pitches because they may not be throwing max effort pitches because they don't have to.
Innings history. Young pitchers often have never had to pitch the 200+ innings a typical major league starter is asked to throw. Add to this that huge innings jumps season to season are tied to increased injury risk, and you often see young pitchers on innings limits early in their career. On a team with playoff aspirations(like the Mets), this can manifest itself via low pitch counts in early games.
Ineffectiveness late in games. It's well established that most pitchers are less effective their third time through a batting order and even less effective their fourth time through. Thus it's becoming more common for teams to pull starters after they've faced just 18 hitters (around 5-6 innings). This can lead to lower pitch counts and lower innings totals over the course of the season.
So while the Mets (or reporters covering the Mets) may be using pitch count as a test, it's not the only one. However, if they are applying it, there are reasons they'd be doing so. The Mets young rotation is a significant asset to their team. It took them to the World Series last year and most of their young hurlers have three things that make them incredibly valuable: age (they're all quite young), talent (they're all quite good), and team control (they're all under contract for the next several years). Because they are such significant assets, it makes a great deal of sense for the Mets to take care of them in the most effective way possible. If they think that's by only allowing some of them to throw 80 pitches a game, then that's what they'll do.