I think someone just made up 300Mm as a max range, probably crossing some wire with the one that remembered that 300km is the (rounded) distance traveled by a photon in a vacuum in one second.
Assuming it's a neutral particle beam (even if the neutral particles are bullshit like "nadions"), in a vacuum, thermal bloom effects should eventually widen the beam. Of course, if it's a plasma (ha ha ha), it should last all of about a hundred feet...
But even for a NPB, the speed of light should have some significant effects on effective weapons range: if phasers move at .1c, as a target at 300Mm, you have nine seconds to see it and dodge it.
If they move at .9c (and it is unlikely that a starship can trivially spin up kilograms of mass to .9c), at 300Mm, you still have a ninth of a second to see it and dodge it.
If they're 1c weapons, of course, they cannot be seen at all until they strike and hence cannot be dodged except by dumb luck. They would also be coherent light, should not be called phasers, and would be very long-ranged.
So, anyway, the speed of the phaser bolt should provide a max effective range against an agile target of 100Mm to a ceiling of, oh, say 600Mm, where in principle it would be very hard to accelerate something fast enough that it could not be seen coming. 300Mm actually sounds like a pretty good limit to being able to hit a ship that is capable of moving itself out of the way in a second or two.
So even though it was totally pulled out of thin air it's not entirely wrong.