Relevant notes and citations provided to TED by Natasha Hurley-Walker.
"... too vast to be explored by spaceship."
According to the best measurements of the Cosmic Microwave Background so far (Planck Collaboration 2016), the Universe is 13.8 billion years old, is expanding, and that expansion is accelerating. As we have known for centuries, light travels at a finite speed, and this speed cannot be exceeded. Even if we could construct a near-light-speed spaceship, we could never explore the visible Universe before it expanded beyond our reach. Even our own galaxy, the Milky Way, would take 100,000 years to cross, in such a ship.
"... hundreds of billions of stars."
Current estimates are that the Milky Way has around 100 to 400 billion stars (see here for a nice primer on why we don't really know exactly). The Gaia mission has mapped over a billion of these, and even measured their motions, allowing them to predict how they will move in the future.
"... a supercomputer here in Perth."
Specifically, the Pawsey Supercomputing Centre
For a longer version of this composite time-lapse, see here.
Bryan Gaensler's Extreme Cosmos provides a fun and engaging tour of some of these extremely energetic objects (and many more).
"... the universe went from neutral to ionized."
The First Galaxies - Theoretical Predictions and Observational Clues (Springer, 2012) has a considerable amount of information on these first luminous objects. You can read the chapter introducing the way astronomers hope to detect it here
"... the beginning of time itself."
Sticklers for detail (myself included) will note that of course, time has already begun when the first stars switched on. We think this occurred anywhere between 100 and 750 million years after the Big Bang -- the uncertainty is why we're trying to find this signal, because it's the last great unknown in the history of our universe. I used this phrasing because without those stars, the universe would just be a big expanding cloud of hydrogen and helium, and by now it would be quite cold and completely dead. So while it's not truly "the beginning of time," it's the beginning of history: of life, of you, me and everything else.