So, is a light-year some kind of diet plan or whatever? The speed of light and space

To answer the titular question in a word: no.

So then, what exactly is a light-year and what does it have to do with the speed of light?

A light-year is the amount of distance that a given light wave (light, radio waves, X-rays) travels in a year. Because the speed of light is approximately 300,000 meters per second, a light wave travels around 9,500,000,000,000 (9.5 trillion) kilometers over the course of a year. Light-years are useful for measuring distances outside of our solar system; a light year is 63,000 times greater than the distance from the Earth to the Sun. Source

Why are you hovering? Get away, mouse!
Our galaxy, the Milky Way, is about 100,000 light-years in diameter. Image

What bearing does the speed of light have on astronomy?

Because light takes a long time to cover the massive distances between the Earth and the stars (the nearest star, Alpha Centauri, is 4.4 light-years away), the light that reaches us here on Earth was produced long before it found its way to Earth. For example, when we see light from a star that is a million light-years away, we are actually viewing that star as it appeared one million years ago – it took the light produced a million years ago that much time to cover the massive distance between the star and our telescopes. Because of this, we are able to observe how the universe appeared in its early stages. When we view objects that are 14 billion light years away, the delay in the light’s arrival provides astronomers with a window into the past.

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