Well, because it takes that long for your aerobic (oxygen using) system to wake up. First couple of minutes there are two other systems that work for you, otherwise you wouldn’t be able to move your muscles at all! This is called your ANEROBIC (without oxygen) phase.
The first part of coming up is using the creatine phosphate (CP), stored in the muscle, and it lasts about long enough to take you through the job you need to immediately do [after the ball is snapped, when you need to jump up off the couch and run out the door, whatever] for the first 30 seconds or so.
OK, 30 seconds, that’s depleted for now. The next system that takes over is glycolysis that takes juuusst about the time you run out of your CP and you’re breaking down carbohydrates stored in your body. You won’t run out of this stuff but it starts building up acid in your muscles and things start to hurt.
This glucose transformation lasts about 90 seconds or so and FINALLY, FINALLY your oxidative (breathing oxygen, if you will) starts to take over and you just begin to be able to fire your muscle cells with the oxygen pathway. This is called the AEROBIC pathway and is able to sustain you through the rest of your swim, run, or whatever you started out doing.
To get there you had to 1) raise your heart rate and 2) start getting oxygen into your blood. This is a rather frantic time for your body so your perception of what’s happening is that you’re really breathing hard and you don’t have enough of anything to even get through the first mile, f’cryin’ out loud.
About five minutes into this whole thing you’re going along just fine but you’ve adjusted to the oxygen part of your cycle and are fairly comfortable with being ‘somewhat uncomfortable’. Welcome to the sustainable part of endurance sports, sport!