Notice how the two verses are parallel:

I was staring at the sky, just lookin' for a star to pray on or wish on or something like that

I went crazy again today, lookin' for a strand to climb, lookin' for a little hope.

I guess staring at the sky makes you crazy, and Fiona isn't the first one to wish upon a star for a little hope.

I was havin' a sweet fix of a daydream of a boy whose reality, I knew, was a hopeless to be had =
Baby said he couldn't stay, wouldn't put his lips to mine and a fail to kiss is a fail to cope

Fixated on a boy who won't kiss you? Seems he is failing to cope, and this reality is hopeless.

But then the dove of hope began its downward slope, and I believed for a moment that my chances were approaching to be grabbed

I said honey, I don't feel so good, don't feel justified. Come on, put a little love here in my void.

Chances of happiness, of a void-filling love.

But as it came down near, so did a weary tear = He said it's all in your head and I said so's everything, but he didn't get it

How wearying it is to explain youself to someone who just doesn't get it?

I thought it was a bird but it was just a paper bag = I thought he was a man but he was just a little boy

Self-explanatory.

Just had to get it out of my system.