How could the universe be a few light years across just 1 second after the big bang? I was skimming through Neil deGrasse Tyson's book Astrophysics for People in a Hurry when I noticed he says something curious on page 26:

"By now, one second of time has passed [since the big bang]. The universe has grown a few light years across."

How could the universe have expanded to a few light years in diameter in just one second if nothing can travel than light? By definition of a light year, wouldn't take a 1+ years for the universe to get this big?
Your question hinges on the misconception that nothing can travel faster than light. No physical object can move faster than light, but plenty of non-physical things can. For example, take the straight line $y = mx + c$. If one varies $c$, the x-intercept of the line also changes. It's possible that this x-intercept moves faster than light for a suitably small $m$. The caveat is that this intercept is a mathematical point, and it cannot be used to transfer information. Relativity isn't violated.

In the case of the universe's expansion, the physical objects in the universe aren't moving faster than light, but they can appear to do so because space is expanding. Let's take Alice and Bob, separated by one meter, and let the space between them expand at some rate. There's no speed-of-light limit on how fast this rate can be. If space expands by a billion meters per second, then one second later, Alice and Bob would be separated by a billion and one meters, but they wouldn't have traveled faster than the speed of light since from their own point of view, they're both stationary.

In the same way, our best guess of the current universe is that the expansion is speeding up. A very far away object might eventually appear to recede from us faster than the speed of light. Relativity remains unviolated, although it means that we'll eventually be causally disconnected from these faraway objects (that is, we cannot affect or even observe them, and vice versa). Something of this sort - exponentially-increasing expansion - happened in the very early universe.