Simple example -- I make LED lighting for fun and profit. I adopted the arduino platform years ago because while I could spend the time to make sure I'm laying out all the components properly (I've done boards for lots of other microcontrollers), I hadn't done USB programming before. So instead of needing to find a partner who could write me an entire bootloader system for doing USB in circuit programming, I can just copy the arduino design, add my secret sauce, and redistribute it.
Or, more recently I've been migrating to ARM. I know zilch about ARM processors, but with the mbed platform I could easily make my light into a USB MIDI device, a DMX platform, and so on. If I had to learn myself how to do all that, it would be impossibly time consuming and I'd never get anywhere.