From a million light years away, it’s a simple answer – the beam spreads out and gets fainter and fainter in any particular bit of space it’s passing through. Just like shining a flashlight on something nearby shows a much brighter and tighter beam than shining it on something far away. So if the source suddenly turned off, we’d see the final bit of its light (very faint and spread out) one million years later.
If the source is much, much farther away (well beyond the Milky Way’s gravitational influence), then not only does the beam get dispersed, it gradually gets redder and redder and if it started far enough away, the light moves to the infrared and becomes invisible to the eye. And if it’s a source at the edge of a black hole, it’s intensity goes down to essentially undetectable levels and it gets redder and redder as it travels.
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