If you've never heard 'I'm bored!' you're probably not a parent yet, or your child must be under 4 years of age! It's a classic, it never gets old!
What do kids mean when they say it? They aren't TRYING to annoy you (we assume, but some might disagree with this). Could they mean 'I'm tired, I'm restless, I feel a bit funny, I need extra stimulation and I miss playing with you!'? Could they mean 'I've recently developed new skills that I want to try out'? Could it mean 'My interests have changed and I'm lost for what to do next'?
I'm bored is the best way they have to describe what they are feeling. They don't have the same grasp on language as an older child or adult does so they are doing the best they can.
Next time your child says 'I'm bored' why not start a conversation? Throw out some suggestions ( you can find HEAPS here) and LISTEN to what they say. Ask open ended questions (ones that can't be answered with a yes or no) to find out if there is something they want to do, but don't quite know it yet. Would they like to do something together or alone? Inside or outside? Something crafty and creative or maybe something that is challenging and requires extra concentration? Maybe they just want to have a chat and a cuddle and not think about doing things for a little while.
And if all else fails feel free to remind them of all the household chores and ask them to take a pick of which one they'd like to help with!