For the same reason all living things reproduce and for one more reason - because just maybe we can. If there is a lifeless place on Earth where life may take hold it it does; as spores, and seeds, and whatever else finds its way there they bring life and new generations with them wherever possible. Wherever nearly possible then then replication with variation and attrition will almost surely make it so eventually.Why should we try to change that inevitable extinction?
If we can find lifeless places beyond our Earth and Sun on which life may thrive but where now there is none, would it not be a good thing to help life find its way there? Maybe a barren world with conditions to support life but lacking the conditions to form on its own, would it not be good to help life to find its way there? Sending life from Earth there is far beyond us now, but not unimaginable. As hard as it would be it would be easier than sending humans though and for each world we hoped to one-day have humans on we might more easily seed primitive life on many times as many.
In that way we could act as seeds for Earth and life which grew here where nothing else can. While we might not be able to make the journey ourselves in any direct way, we could at least provide some possibility that the biodiversity that has developed here does not end here and thus serve it in some way more than anything else living now can. Otherwise we end without doing this and serve nothing beyond ourselves and even then serve ourselves poorly.
I'm not saying we're doing a very good job of living up to that potential mind you. Its something we can offer that nothing else living on Earth as yet can; it would be at once our redemption and Earth's salvation if we were to realize it. Were we as a species to end without living up to it we would be dead without redemption and could not be quite certain such a potential would arise again nor if it were whether it would likewise be squandered. Our failure in this would stand as the most lasting fact of our history. While we can still do something more - something worthwhile in a really big way - I don't think it would be right not to try.