The movie begins with Jane and Michael getting lost after chasing a kite away (which happens off screen), and ends with the whole Banks' Family, flying a kite together.
In other words, the movie begins with chaos and disorder, and ends with control and family happiness. That's what kite flying symbolizes: it's a fun and simple thing a family can do together. It doesn't require anything fancy or a lot of money, but it's still cool. As George sings in the song, "Let's Go Fly a Kite":
GEORGE: With tuppence for paper and strings
You can have your own set of wings
With your feet on the ground
You're a bird in flight
With your fist holding tight
To the string of your kite
(There's the tuppence again—happiness with simplicity, just a little money required).
Also, in order to make the tail for the kite, Mrs. Banks uses her "Votes for Women" sash. This seems to be implying that she should lay off the activism and spend more time with her kids—which is sort of disputable, given how important gaining the right to vote was for women.
Hopefully the next family outing is to a suffragette rally…