The prhase "I can't bring myself..." means exactly what says. It isn't even an idiom, really. It means you cannot carry yourself to that task. In order to do the task, it would be necessary first that you be there to perform it. Alas, you are unable to do even that, and if you can't even be there then you certainly can't do it.
Other ways to express the same idea:
I can't face doing that
I'm just not up to (doing) that
I can't even think about doing that
I can't make myself do that