This sentence sounds like informal, spoken English. I would make some slight changes to yield the following:
I was told that I would need to substitute for you until September 25th, which means that they are all yours starting tomorrow.
This sentence makes sense if today is September 25th. I imagine that one teacher, who was acting as a substitute teacher, is speaking to the returning, regular teacher on September 25th. The substitute teacher is saying "the students are your responsibility (once again), starting tomorrow."