I seems a bit like you're assuming the big bang occurred at a place; that it was an explosion of tightly packed stuff into a pre-existing, empty space. That's not what the big bang model says.
Instead, the big bang was a rappid expansion of space. Whether that is all of space, or just some local pocket of space is still an open question in GR cosmology, but either way we start with much, much less space than we have today, and, as a result, the universe was a much hotter, much more dense place. It was so hot dense, in fact, that light was only able to travel was basically zero.
Then, for some reason, the space between stuff began to grow. For people who adhere to hyperflationary theory, it grew so quickly that space expanded at many, many times the speed of light. This would have taken those light travel zones from being nanometres across to be several meters across. The distance light can travel in this new, expanded universe has grown to be 10s of centimetres, and the 10s of centimetres across region that has the location of present day Earth at its centre is our observable universe.
For those that don't adhere to hyperinflation, this same thing happened, but more slowly, meaning the 10s of centimetres across region represents a larger chunk of the stuff at the beginning.
The universe continues to expand at such a rate that the horizon region is shoved away from us at somewhere near the speed of light. Maybe a little more, maybe a little less. Regardless, we never see much beyond those original few 10s of centimetres. Over the billions of years that have followed, the distance to that horizon has grown to some 40+ billion light years away. And remember that that's the stuff at the distance that the universe became opaque at, so it must looks like a glowing haze (because that's all it was at the time that light left it).
And, of course, the expansion of the universe has been accelerating for several billion years now, so the horizon actually is expanding away from us at greater than the speed of light today. So things that were once within our field of view aren't anymore (though, to date, the stuff that we can't see anymore is just glowing hazy gas).
So, in short, we've never been able to see everything, and we see a little bit less every day.
