This is where you’ve gone wrong. To be fair, it’s where most people go wrong.
A lot of people get the impression that the Big Bang was matter exploding out into an already existing empty space. That’s wrong. This is what also leads to your incorrect supposition that there is a place you could be where you could see the exploding matter coming towards you.
To really understand this, imagine that you have a flag that you put at some place in the universe. It’s a magic flag that never, ever moves from wherever you put it. It’s going to stay in that spot forever.
Now imagine that you have a second magic flag, that you place somewhere else in the universe. This flag also never, ever moves. Wherever you put it, that’s where it is, forever.
Now here’s the fun part. Even though neither one of those flags can move, they are both moving away from each other. They aren’t actually moving. The space between them is getting bigger. It doesn’t matter where you put those two magic flags. Wherever you put them, they keep getting further apart, even though neither flag is moving.
Now if you rewind that backwards in time, the flags get closer together. They all keep getting closer and closer and closer until finally they all squish together in a single point. That flag right next to you, it ends up squished in that single point. That other flag you plopped down on the other side of Alpha Centauri, that flag also gets squished down to the same point.
So where did the Big Bang originate? Right here. Where you are now. And way over there in that other galaxy. They were all the same point. There is no “outside”. All of the outsides are at that same point.
Admittedly it’s kinda trippy, but wrapping your head around that is the key to understanding what the Big Bang really was. It wasn’t just matter exploding out into a space that already existed. Space itself started from a single point and expanded outward.
Initially, all of the matter was squished together so tightly that the heat of it all being squished together prevented atoms from even forming. Eventually, everything cooled down enough that atoms could form, mostly hydrogen but some helium too, I believe. If you somehow existed at this time, you wouldn’t be able to “see” anything because there weren’t any stars or anything to produce light yet. And no carbon or other heavier elements to make a “you” out of either. Eventually the hydrogen and helium gas molecules clumped together and once those clumps got big enough, the pressure from all of the gas molecules being squished together (from gravity) caused fusion to start, and now you had stars. Now you finally had some light that you could see with.
Except you couldn’t exist yet to see anything, because the heavier elements that you are made out of didn’t exist yet.
Those early stars created the heavier elements through fusion, and when those stars got old and died, the bigger ones went all kablooey (how’s that for a technical term) and scattered their heavier elements all around out into the nearby universe. Those elements helped form newer stars and newer planets, and some of those planets (like Earth) ended up with lots of nice heavy elements to make interesting stuff out of. Like you. And now here you are.