There are a number of ritual mitzvot - lulav, shema, megillah, chanukah candles, prayer, bedikat chametz - regarding which you may not be allowed to eat in the morning/evening before performing the mitzvah. This is meant to encourage you to do the mitzvot ASAP and not delay. But for the mitzvah of shofar in particular, this prohibition on eating may not exist, or else may be weaker than for other mitzvot. Why?
I would like to add an explanation to the ones in the above-linked article. For most normal mitzvot, the mitzvah is to do a particular action. For example, once you have finished "taking" the lulav and "talking of" the shema, you have "gotten these mitzvot over with" and there is no more obligation. To discourage unnecessarily delays in "getting them over with", you are not supposed to eat beforehand. (The phrase "get them over with" makes it sound like mitzvot are an annoying burden, which they shouldn't be, but it accurately conveys the attitude of "when it's done, it's done".)
Shofar is different. The Torah does not say "you shall blow the shofar", but rather "you shall have a day of shofar-blowing". The mitzvah of shofar is not to perform a particular action, but to create a certain mood for the entire day. No matter when you blow the shofar, the exact same day becomes imbued with the essence of shofar. In a sense, blowing the shofar is not the mitzvah but just a means to the mitzvah. Therefore, it is less crucial that it be done at the first possible opportunity.
There is another possible approach which follows from the same basic ideas. Rosh Hashanah is a "day of shofar-blowing", but it is also a day characterized by other things, such as eating a yom-tov meal. There is no reason to privilege one aspect of the day over another, so you are allowed to eat first as well as blow the shofar first.