There are two directions satellite to ground and ground to satellite, and they operate slightly differently.
From the ground you can think of an array of satellites like a monitor showing a document. As you read every word is being displayed at the same time in the same color, but because the light is coming from different locations you can focus on a little area and ignore the rest of the screen.
A traditional satellite dish uses a similar idea to just look at a single satellite at a time and ignore the rest of the sky, this is why you need to aim them. What starkink does differently is using a phased array to let it change focus without physically moving the dish, which is really really fast and lets them track satellites in LEO.
Just like cellphones it can hand off from satellite to satellite plenty fast to keep up with the speeds involved. As soon as you can handle 1 transistion just repeat the same thing millions of times, computers are great at repetitive tasks.
As to throughput, it’s supposed to be fast enough to watch 4K video which isn’t gigabit speeds but is vastly better than millions of people are stuck with.