Memory and the Past
I will always remember when the stars fell down around me and lifted me up above the George Washington Bridge. (2)
The very first sentence tells us that Cassie is sharing a memory. When you "remember" something, it's in the past.
[The quilt around the borders of the pages] 
Quilts aren't exactly modern or new-fangled, you know? They remind us of the past.
[Laundry lines] (3-4)
You're not going to find too many laundry lines on the rooftops of New York City these days. That's one way we know the story is set in the past.
The bridge was my most prized possession. (6)
We're still in the past here, in Cassie's memories. The bridge was her most prized possession. That suggests it isn't her most prized possession anymore.
Daddy said that the George Washington Bridge […] opened in 1931, on the very day I was born. (7)
Hey, here's the clue that helps us know when the story takes place. Let's do the math: Cassie would've been eight years old in 1939.
Daddy worked on that bridge, hoisting cables. Since then, I've wanted that bridge to be mine. Now I have claimed it. (8-9)
Now, wait a second. Something crazy happens here. Daddy worked on the bridge in the past. Now—now—Cassie has claimed it. We're shifting tenses here, people.
Me, Cassie Louise Lightfoot, only eight years old and in the third grade, and I can fly. (10)
Yep, that's the present tense all right. We have officially traveled back in time to 1939-ish.
But still he can't join the union because Grandpa wasn't a member. (13)
Unions used Mr. Lightfoot's heritage as an excuse to not admit him into the union. Gross.
And Mommy won't cry all winter when he goes to look for work and doesn't come home. And Mommy can laugh and sleep late like Mrs. Honey. (16-17)
In her fantasy, Cassie's mom doesn't cry. That suggests she did cry in real life…which is terribly sad.
Tonight we're going up to Tar Beach. Mommy is roasting peanuts and frying chicken, and Daddy will bring home a watermelon. (21)
Wait a second…Daddy's back. And Cassie's telling us about the future, telling us about going to Tar Beach later tonight.