Sadly, the most reasonable thing to do at this point is to find another school in another country, that's how bad the process is. If you still have any interest in France after this, it will be easier to immigrate later, maybe on a work visa or using some EU law dispositions or even try your luck again with a French business school after collecting additional experience abroad rather than fight for a student visa now.
First, it's perfectly legal for the consulate not to provide any explanation when they refuse a student visa. Schengen refusals have to be motivated, family visa refusals have to be motivated, but not student visa refusals. And, unlike for, say, family visas, consulates have a lot of discretion in granting the visa or not (and that's not a coincidence: if you applied for a visa to which you are entitled based on your situation, they would have to explain why they think you don't qualify). You have to submit a cover letter and a resume and the consulate is legally empowered to evaluate that independently, an acceptance letter from a French university or higher education institution is no guarantee that you will get a visa.
In principle, you always have the right to appeal if you think the refusal was illegal. In practice, you can of course ask the consulate what their reason was and they are allowed to tell you... but they don't have to do it if they don't want to or to provide any justification in writing. And it's obviously very difficult to appeal effectively if you are not entitled to a visa in the first place and you don't know why it was refused!
Now, the appeal procedure itself also makes things very difficult, even if you have strong arguments on paper (which is probably not your case, as we just established). You can lodge a recours gracieux asking the consulate to reconsider its decision and you could probably try that, as it does not cost anything. On the other hand, for a student visa, it will almost certainly be ignored. You do need to find some cogent arguments and to write in French, of course.
At the same time, if you really want to exhaust all legal means, you will want to make a recours contentieux, i.e. go to court. But you can't do that directly, you first need to send your appeal to the Commission de recours contre les refus de visa d'entrée en France. In practice, it's not very helpful as it just confirms 98% of the decisions it evaluates and struggles to give an opinion within the two months the law prescribes. But you don't have a choice: you must first formulate an appeal in front the commission and wait for a two-month delay to elapse before you can actually go to court.
And then the real fun starts. All visa cases have to be treated by one specific court, the tribunal administratif in Nantes. Once you have lodged your appeal there, the foreign ministry has a certain amount time to present its arguments (something like 4 or 6 months, not sure anymore). It routinely ignores this delay but the court still waits for it. And it's overloaded anyway, what I read suggests that visa cases usually take one and a half year to reach a decision (and anecdotal evidence from friends and relatives bears this out).
This delay is bad enough in itself but it's even worse for you, as your admission letter probably won't be worth much in two years from now and you personal project will be less plausible if you just sit idly waiting for the decision. This means that even if you would manage to get the court to void the last refusal, the consulate would be perfectly justified in refusing again based on your new situation. And the odds are not good: The tribunal only finds in favour of the applicant in something like 20% of the cases (sorry, lost the source for that number) and that's only from the fraction of cases that are brought to its attention, i.e. the people with the strongest legal case for their appeal, the vast majority of unsuccessful visa applicants understandably don't bother.
And the icing on the cake is that the state can pull its decision at any time. In theory, it's not a stupid idea: If the state reviews your case and finds the original decision was wrong, it can correct things immediately. But in practice, it can be bad for you because the TA in Nantesd does not award the costs in this case (even though it should). Worse, I actually know someone who brought a case against a similar decision (it was not about a visa but a related procedure), was (probably) about to win it and then, after several months, the ministry suddenly reversed its original decision... and issued a new refusal, on another ground! She was stuck with her lawyer fees for the first appeal and had to start a new one from scratch. She eventually succeeded but that's the kind of things you have to brace for and, for a student visa, it probably does not make sense to put yourself through this.