It's not fixed at June 1st. That date is just a rough guideline as to when a player may be called up. The date when the player arrives can often be just before or after the June 1st date, (e.g. Ryan Braun was brought up near the end of May three years ago) and the goal is to push back that player's arbitration eligibility by one year. By delaying that player's start to his MLB career, teams hope to avoid making him "Super-Two" eligibile. Once a player has three years of MLB service he is eligible for arbitration and a likely pay raise. However, a "Super-Two" eligible player is eligible for arbitration with less than three years based on the following guidelines in the collective labour agreement:
"[A] player can be classified as a "Super Two" and be eligible for arbitration with less than three years of service. A player with at least two but less than three years of Major League service shall be eligible for salary arbitration if he has accumulated at least 86 days of service during the immediately preceding season and he ranks in the top 17 percent in total service in the class of Players who have at least two but less than three years of Major League service, however accumulated, but with at least 86 days of service accumulated during the immediately preceding season."
So the teams delay the start of a player's MLB season by roughly 2-3 months using a calculation based on the above mentioned clause. For each player it will be a bit different hence the arrival date usually being late May and into June.
"Trying to hit him was like trying to drink coffee with a fork." - Willie Stargell on Sandy Koufax