The bolded part in your post is why it is begging the question--in your assertion, the premise assumes the truth of the conclusion and therefore offers little support for its veracity. I'll give another example to try to elucidate how it works:
Water seeks its own level because it is a fluid.
As "reasonable" as this sounds, it begs the question because the premise (water is a fluid) assumes the conclusion (water seeks its own level). A fluid generally has the properties of a compound that will seek its own level over time (chemists, don't slam me for this--glass was just de-classified as a liquid and is now either a solid or something else). So the conclusion hasn't been adequately supported with unique premises, premises that are almost completely independent of the conclusion and/or lead directly to its veracity.
I think that if you can make a statement into a circular argument by swapping the premise and conclusion, question begging is likely to be involved. In the Rutgers example, the circularity of which I refer would look like this:
Example 1
Q. Why isn't Rutgers the best team in the B1G?
A. Because they don't win many games.
or,
Example 2
Q. Why doesn't Rutgers win many games?
A. Because they aren't the best team in the B1G.
This sort of breakdown is an indication (at least to me) that begging the question is involved, inasmuch as one is linking a conclusion to a premise that is essentially the same statement as the conclusion, and therefore offers no support for it. There are a lot of fallacies and rules that go into breaking down arguments, and it can be tortuous (as many or all worthwhile fields of study are), but question begging becomes more obvious when you hear somebody make a statement for which the premise assumes some or all of the content of the conclusion. All that said, I think there is a better example than my Rutgers/B1G example, but I'm rusty after all these years. And rest assured that if a guy like Skillet initially thought it wasn't a good example, but then nuanced his way through it to see that it works, I could've come up with a better example. He is capable of hairsplitting with the best of them, and equally adept at explaining his breakdowns, but most folks aren't in his league. Maybe someone else in here can answer your question if my answer is either unsatisfactory, wrong, or both.