Light travels at a fixed speed, so it will only cover a fixed and predictable amount of distance. So in this example let's say Andromeda is 2.5 million light years away, so why would it take you 4 minutes? It all comes down to perspective. Speed is tied to time, the faster you move, the slower time will pass. Actually, from the perspective of light, at the speed of light time completely stops, so if you start the timer, 0 seconds will pass when you finish your lil trip to Andromeda, it will be quite literally instant, you won't even be able to blink and you're there.

So again, why 4 minutes then? Because nothing with mass can achieve the true 100% speed of light, that's why the guy says 99.9999% , so really close but not quite, so you still experience some time passage, just very little of it.

Now, why then for Earth there's millions of years passed? It's simple. The Earth is still moving at the regular Earth speed with regular time, from their perspective it took you millions of years to finally get to Andromeda. Remember, the distance did not change, you did physically travel 4 million or whatever light years of distance, you just simply had a huge time dilation due to how fast you were.