How can light reach us from 14 billion light years away? One thing that I can't quite wrap my head around is how light is traveling to Earth from 14 billion light years away, aka the beginning of the universe.

The way I see it, the universe itself was very small 14 billion years ago and has been steadily accelerating in size since.  So the photons that left a star that existed 14 billion years ago couldn't travel for 14 billion years because the universe itself was probably only a few million light years total in diameter.  The universe is not expanding faster than the speed of light, so that photon should have hit the end of the universe before now.

What piece of this puzzle am I missing?
If a star 300,000 km away from you explodes, it'll normally take the radiation from that explosion one second to start vaporizing you.
If the space between you and that exploding star inflates to 2.6X10^23km (28 billion ly) one half second after the explosion, the radiation will take 14 billion years, plus a half second, to reach you, and won't vaporize you.
Of course the expansion need not be instantaneous. You can plug in about any function you want there. The important thing is not the distance between you and the star at the time of explosion, but the total distance the photon must travel on its path to you. That distance must take into account the expansion of space between the emitted radiation and you, the observer.