Okay, let's start with the basics. The Big Bang was not like an explosion in space from which spewed all matter in the universe. The Big Bang was a moment in time. We have this thing called a spacetime metric. I won't bore you with the details, but essentially it is the equation we use to describe all of the geometry in the universe. It includes all the dimensions and the dips and bumps and warpings that the gravity of massive objects imparts on them. In this metric, there is something called a scale factor, $a$, this scale factor is something that describes the expansion of the universe. It is multiplied by the spatial dimensions. In our metric, any spatial distance between two points is the distance that we would measure with a ruler today, in the present. To that end, we define $a=1$ in the present. Because space is expanding and the distance between two points grows as time goes on, we know that a distance we measure today would be smaller in the past. Therefore, in the past $a<1$. For example, today we might measure the distance between two points to be 1 meter, but back when $a$ was $0.5$, the distance between those same two points would have been measured as half a meter. Stick with me, I'm getting to the point. The Big Bang is defined as the moment of time when $a=0$. That means the distance between two points in space was zero. It does not mean that everything was all in the same point, it just means that a ruler would measure the physical separation between two distinct points in space as zero. So you see, the Big Bang was a moment in time (like yesterday), not an event that happened somewhere.
Okay, so far so good? Now, when matter came into existence, it was distributed more or less equally all throughout the universe. It did not all start at the same point in space, it started everywhere. It's understandably a bit confusing given that at some time, the distance between any two points was zero, but there is an important distinction here. Think about that now. Matter started off at different positions. As space expanded, more distance was added between two adjacent points, but the matter occupying those points never actually moved from them. 1 meter apart became 2 meters, 2 became 4, 4 became 8, and so on. The way space expands is that you get a certain amount of distance added each second to a given distance. For instance (and note, this number is not the real number, it is for teaching purposes only), for every 1 meter of distance between two points, you might have 2 centimeters added every second. If you think about it, that means that something 20 billion meters away from a given point would have about 400 million meters of distance added between them in one second. Technically, that means any object at the second point would be getting farther away from the first point at a rate faster than the speed of light. But this is okay! The object at the distant point is not actually moving from that point and other objects near it wouldn't see it going faster than light or anything like that. The speed of light is not violated.
Now let's talk about how looking out at the universe is looking into the past and how the universe is 90 billion light years across while only being 13.8 billion years old. Remember I told you that matter did not start all in the same place, right? Well imagine that about 13.7 billion years ago, something far away from where we now are emitted a beam of light in our direction. When I say far away, I mean it was probably a couple million light years away (the early universe expanded quickly). You might think that the light should then only take a couple million years to reach our position, but no! As the light travelled, space expanded beneath it. Each second that little beam of light would cover 300 million meters, but each second more distance would be added in front of it and behind it. After a million years, that light beam was (frustratingly) only a bit closer to reaching our position, say 1.999 million light years to go instead of 2 million (progress!). Moreover, it was now more than one million light years away from its point of origin because space was expanding behind it. Over the next 13.7 billion years, it inched ever closer. It was able to shorten the distance by more every year because every year there was less space between it and us. Eventually, it reached us. It started 2 million light years away from us, travelled 13.7 billion light years, and amazingly, it is now about 46 billion light years away from its point of origin. That's space expansion for you. That also means that when we view this beam of light, we are looking at something as it was 13.7 billion years ago; something that is now about 46 billion light years away.
Are we moving faster than light? No. Did those other galaxies have to move faster than light to get 45 billion light years away in 13.5 billion years? No, they just sat in space and the distance increased on its own. I hope this answers all of your questions. Cosmology is an interesting subject and often it's more wild than popular science media would lead you to believe.