Q:

What is the meaning of "The rate I'm going"?

In Angelina Jolie-directed movie By The Sea (2015), Vanessa taking sun bath and her husband writer, Roland, asked this question to her.

Roland: Wanna go for a drive?
Vanessa: I thought we agreed we weren't going anywhere  until you
  wrote.
Roland: The rate I'm going... Let's go for a drive.

A:

Sometimes words that are obvious or understood in the context can be missed out.  This is called ellipsis.
http://dictionary.cambridge.org/grammar/british-grammar/about-words-clauses-and-sentences/ellipsis
"The rate I'm going", in this context, means "at the rate I'm going", i.e. "at the speed I'm going at", "at the speed I'm writing".
The implication is that he's writing so slowly, or progressing with his writing so slowly, that it will be a very long time before they can go for a drive (based on the principle originally set out)  so they may as well set that rule aside and just go for a drive anyway.
The meaning is along these lines:

Vanessa: I thought we agreed we weren't going anywhere until you
  wrote.
Roland: [At] the rate I'm going, [that's going to take forever!]  Let's go for a drive.

