Someone else thinking my writing is good enough to publish (or good enough for anything other than the bin) validates that writing. It also validates my writing ambitions. It makes me feel a little more confident.
It's stupid: my writing being published doesn't mean it's better than my unpublished work. It doesn't automatically transform those 300 words into something good. Besides, there are thousands of mediocre writers whose work is published in some form. Many of them make a lot of money from terrible books. I know all of this, so why should such a negligible achievement make me feel validated?
I suppose it's evidence; one or two people, at least, think my words are worth reading. Or at least worth filling a few inches of blank paper. If one or two people think my writing is okay, perhaps more people will think it is okay. Perhaps it is worth taking the time to write and the trouble to submit work.
That's why validation matters: it gives writers the extra iota of confidence they need to keep going. So much of writing is a solitary struggle that it's good (even essential) to be reminded that it's not a waste of time. I'm pleased to be published in my local newspaper not because it's the culmination of a dream, but because it encourages me to keep pursuing my dreams.