I had no idea what this meant when I saw it, but now that I know what it means, I happen to like it.
So what does it mean?
Essentially this means "I'll make sure...". Here are some examples:
De mi cuenta corre que hables más como un nativo
I'll make sure you speak more like a native
De mi cuenta corre que seas muy feliz
I'll make sure you're very happy
This can also be expressed with....
Yo me encargo de que
Yo me encargo de que consigas un lugar donde quedarte
I'll make sure you have a place to stay
I think is a great piece of Spanish, I can't wait to use it.
¡Espero que disfrutes!