Hello,
I was an undergraduate international student in USA. I do not know the complete answer for your case but I can tell you what I have learned so far.
First, since the MSF you are looking at is a one year program, the duration of the I-20 will be one year. That means that you can stay in the USA legally until a certain date. It is important to stress that I-20 is by far more important document than a visa. Believe it or not, for international students the visa itself is only a document that allows them to enter in the USA. But you also have to show your I-20 to the people at the airport. Without it you cannot enter even if you have an F-1 visa. Visa can expire while you are in the States but it means nothing to the international student since he or she has the I-20 that is going to expire in the future. However, if it happens that your visa expires and you leave the States while your I-20 is still valid, you will need to apply again for the visa to enter USA.
Second, there are two ways to continue your stay in the USA. If your grades/gmat/application is well enough, during your studies for MSF you can apply for an MBA. If you get admitted by the school, it will issue you another I-20 for your MBA.
As far as what goes next I am not 100% sure: I imagine that you do not have to go back to your country if you get admitted to the MBA program right after your MSF. I think because you will have I-20 for one year as well as for the next year there is no reason for you to leave the States if you do not want to. I know some international students who have graduated from a college and immediately after went to a Master program. They did not leave the States.
Another way to stay in the US is to apply for an OPT. That should give you the right to live and work in the States for one more year upon graduation (one year is for undergrads, not sure how much it is for master program). I do not think that you can get extra year of OPT after your MBA if you used it right after MSF.
I hope that this did not sound too confusing