In the instruction manual, it says you can increase your creative skills by either playing music, or sewing clothes.
Now, I've used the sewing kit to repair my damaged clothes, and I've also used the craft table to create new clothes, but neither one of these seem to increase my creative skill (because I don't see the blue skill meter come up).
What's the deal?

the meter didnt come up with me either, but their skill still went up.

Ah, I see, that's good!
Thanks for the reassurance, it was rather irritating not knowing if my efforts were actually doing anything.

The fastest way i know of to raise creativity is to use the monkey horn, which is found in the llama temple. Trust me, mine was 3, and i raised it to 10 in less than a Sim day;)

I always make clothes once I get some decent clothing plans unlocked and I always repaired the clothes once I had made a sewing kit.

Making clothes does increase the creativity, but I never noticed when it came to using the sewing kit.

My main method of increasing creativity has always been to craft a Shaker and have my Sims play it. It provides Fun as well as increasing Creativity when it is played.

Make one and then leave it when you go to another location and then when you check there will be one or more in your inventory and the one that you fist made will back where you left it.