The Scarecrow is the first character to join Dorothy and Toto on their journey to see the wizard and claims to not have a brain although right from the beginning he solves problems. This continues throughout the movie and he does the most problem solving and thinking as the group travel towards Oz.
The character of the Scarecrow is important in The Wizard of Oz movie because he is the character with all the brains without him realising it and Dorothy wouldn't have made it far on her journey without him
Here is an example of the Scarecrow using his problem solving skills, this is the scene where he is found by Dorothy and taken down out of the field. scene here
In the clip you can see the Scarecrow being helped down from the pole by Dorothy and using the brain that he says he doesn't have to help solve the problem of getting him down.
The Scarecrow is a character who feels the need to have a brain but doesn't realise that he has 'smarts' and the ability to think right from the beginning.