Isn't the obvious answer, "because, assuming your life isn't unbearably bad, living the next 1,000 years has higher expected utility than not living the next 1,000 years?"

We don't have accurate predictions about what the next 1,000 years are going to look like. Any probability calculation we make will be mostly influenced by our priors; in other words, an optimist would compute a good expected utility while a pessimist would reach the opposite result.

Responses like yours confuse me because they seem to confidently imply that the future will be incredibly boring or something.

I'm saying that if there's nothing impressive about my life in the present or the past, then I'm not one to expect much more out of the future. Some people have a cause or goal and would like to live long enough to see it through--good for them, I say.

I harbor no such vision myself. It's possible that something comes up at a later time and, over the course of 1,000 years (say), it seems rather likely that at some point I'd encounter that feeling. It's equally likely that something unavoidably bad comes up. On balance, I'm indifferent.