Spoiler:
The purple circle does definitely represent Kirby and I'm just as sure that the green rectangle represents Yoshi. The purple rectangle does probably represent the bandit, I'm not sure about the white triangle but the shape suggests that it's a female. Could it be Abby? Now, before we start speculating about what's really going on in this dream, let's analyse those black and white spots inside the shapes. My guess is that they represent the amount of good or evil in that particular person. So what's really going on in this dream? Let's analyse it square by sqaure.
Square #1 Kirby and Yoshi are both happy and all is well but...
Square #2 They meet the bandit who is quite a bad guy judging from the big black dot with only a little bit of white inside. The two white lines with the explosion-like object at the bottom indicates a fight.
Square #3 We see that the bandit has gotten a lot nicer, only a little bit of black remains under the white dot. It also seems that he has teamed up with Kirby and Yoshi. The three friends suddenly meet someone ho seems to be very feminine and with a heart of gold. Also notice the small yellow marks on the new character which could represent some kind of jewelry or possibly some sort of power, a princess perhaps? The picture in the down right corner suggests that this all takes place near water.
Square #4 WOW, suddenly we find ourselves in space with the stars above us and the earth's surface below us, facing someone very dark and evil. Just look at that burning red blob of hatred in this new menace's oval. There's also something very interesting happening in the middle with that s-shaped object, I have no idea what that's supposed to mean though. Another interesting note is that the yellow markings on the triangular person are gone. Why is that?
Square #5 The new evil has transformed into some kind of monster, notice his white oval that is still there. What we see here might very well be the final battle and it seems that the feminine triangle plays a big part here. I have no idea what the lightbulb means but isn't that the X-naut emblem at the bottom? So we're in space with the X-nauts, this makes me wonder about that yellow shining thing above the water in square #3, could it be some sort of space ship?
Square #6 The dream is over and Kirby awakes