When July became December
Their affection fought the cold,
But they couldn't quite remember
What inspired them to go.
And it was beautifully depressing
Like A Streetcar Named Desire.
They were fighting for their love
That started growing tired.

These lyrics are referring to Ryan Ross and Jon Walker leaving the band. They broke up in July, so that's where the lyrics "When July Became December" came from.

The lyrics in this line "But they couldn't quite remember what inspired them to go." is Brendon talking about Ryan and Jon leaving to create a different style of music that they were inspired to do.

"And it was beautifully depressing" means that Brendon was depressed that Ryan and Jon were leaving, but he wasn't entirely because he knew Ryan and Jon were leaving for the best.

The lyrics "They were fighting for their love that started growing tired." mean that maybe Ryan and Jon's love for Panic! At The Disco was starting to pile over them because they were not enjoying the music they were coming up with, and they were getting tired of it so they decided to go their own way and make music as The Young Veins.