I like them, getting a platinum gives me a feeling of personal accomplishment in gaming, kind of like getting 100% in a game. For me it's definitely not about bragging rights, I doubt anyone else other than my 3-year-old would even bother looking at my trophy list.
I also like using the trophy list as a to-do list in gaming for myself. Any game lacking in earned trophies is generally not something I've managed to get my money's worth out of yet, so it reminds me that I still have to spend more time playing it to justify having paid for it.
The only game I ever received a platinum is Criminal Girls invite only. Not sure if I should be proud or ashamed of that milestone, probably both.