Lots of the usual highly educated and very intelligent answers. I have a not-so-smart one, which might be a little more useful.
It’s fairly simple: when a star shines, it sends it’s energy out in every direction. As each little energy packets (photons or neutrinos or powdered donuts, or whatever the star likes to emit) move outward from the star, they obey the Newtonian law of keeping on in the line they started from, unless something affects them.
If you are right next to the star, even though each packet is going in a different direction, a lot of them will hit you, and you’ll have what we scientifically refer to as “a bad time of it.” But the farther away from the star you are, fewer packets hit you, because they are headed in other directions. Since the closest star to us other than that big orange thingy is four light years (approximately) distant, almost none of the energy coming from it is directly headed for us.
You could do a small experiment if you like, to visualize it more easily. Point a flashlight at a wall, while standing next to it. Then back away, and notice that the area that is lit up gets larger, but it doesn’t get lit up as brightly. That’s because the photons being “sprayed” on the wall by the flashlight, are going in a lot of different directions, and not just straight ahead. Use a laser pointer, and you have to back away much further to see the effect.
In short, aside from running into the bits of junk between us and the distant stars, NOTHING happens to the energy coming towards us. It’s just that the vast majority of the energy of each star ISN’T coming towards us.
By the way, the reason why the “sky” is dark, isn’t because the stars haven’t been shining long enough. It’s dark because most of the light coming from each star, is going somewhere other than here.
The OP really had the answer right at the outset, when he referred to most of the local star energy going elsewhere, and not “hitting” us. The energy from all the other stars, is ALSO not hitting much, because most of spacetime is empty.