If the universe is expanding and the speed at which it is expanding is certainly consistently lower than the speed of light, then the light emitted by the birth of the universe must have passed us by a long time ago, right? How is it possible to look back and see the birth of the universe?
Many people struggle with the idea that looking farther out means looking farther back, at least in the temporal sense. Presumably, you already get this point, or at least part of it. Just to make sure we're on the same page though, along with anyone who might stumble across this, let me put it this way:
The light emitted by the birth of the universe comes from… well, everywhere. The early universe was a boiling sea of high energy particles. These particles were packed closely enough to keep light trapped basically right where it was sitting. In other words, the universe was opaque.
However, the universe was expanding, which means the boiling sea of particles was cooling off, as all expanding substances do, and eventually, the whole shebang cooled to the point where light could escape, which it did, from every point in the universe at once.
So let's imagine you're sitting in a spaceship in the middle of all of this, watching the universe become transparent. (With appropriate radiation shielding, or some sort of super-robot body. Use your imagination.) What you'd see in the first instant is all the now-free light rushing at you from the nearby part of the universe, naturally, the light that's had time to reach you. Wait five minutes longer, and you're still seeing light, just light that had to travel farther to get to you. But all this light was emitted at the same time, so you conclude that the light is five minutes old, and you're seeing something that happened five minutes ago.
Hopefully, you see where this is going. Wait around a few billion years, and you're still seeing light from that same event, way back when, but which was emitted so far away that it took all of those several billion years just to reach you. So by virtue of the long unfathomable distances between, you get to look back in time.
The key to your question may be in your understanding of the place in time and space from which the light was emitted. If the light had been emitted immediately, i.e. at the same instant the universe was created as a tiny little speck, then all the light would certainly have raced past us by now. But we're not looking at some cosmic firecracker, the burst of light that was produced by the explosion that started it all, in fact we can't, precisely because the universe was too bright for the light to move around for the first few hundred thousand years. (Well, too energetic, anyway. Hopefully, this makes sense.) By the time the universe was transparent, it was also very big, such that we can look out to long distances at times long ago.
Which would be the whole story, if the universe had stopped expanding once it turned transparent. It did not, however, so the answer is a little more complicated (though if you made sense of the above explanation, you should be in good shape). It turns out that your second assumption–that the universe expands more slowly than light speed–is wrong.
Which might sound like a joke at first, but I assure you I'm serious. Objects are constrained to move at velocities less than light speed, but space itself may stretch as fast as it wishes, over great distances. Of course, if the space between two points is stretching faster than light, then those two points are prevented from ever trading information, and might as well belong to entirely separate universes.
So our universe effectively has some maximum volume, namely, that volume which is expanding away from us more slowly than light speed. It just happens that this volume still encompasses light that was emitted in the earliest years of the universes existence. Presumably, this light will eventually disappear over the edge from our perspective, never to be seen again.
Thanks to Alex McDaniel, David Lai, Shawfeng Dong, Gabe Prochter, Ian Dobbs-Dixon, Jay Strader, Justin Harker, Karrie Gilbert, Kyle Lanclos, Laura Langland-Shula, Lynne Raschke, Marla Geha, Michael Kuhlen, Nick Konidaris and Scott Seagroves