There are a number of situations in which halacha requires us to refrain entirely from talking. For example, you cannot speak between laying the tefillin Shel Rosh and the tefillin Shel Yad. In fact, the Gemara tells us that if someone does speak then, he is unfit to serve as a combat soldier on the battlefield! Likewise, according to some opinions, you cannot speak while checking for chametz on the night before Pesach (בדיקת חמץ).
Another no-talking time is from the time the first set of tekios (shofar blasts) is sounded, until the last set of tekios during Mussaf, a period totalling an hour or two, or even more.
After reciting a brachah on a mitzvah, you must immediately engage in the mitzvah. On Rosh Hashana there are two main sets of shofar sounds referred to in the Gemara as תקיעות דמיושב and תקיעות דמיועמד. The terms imply that the first set is done sitting, but in fact today all of the shofar blowing is done with both the shofar blower and the congregation standing.
When do we recite the brachah on the mitzvah of hearing the shofar? Before the Tekos D’Meyushav, before the Tekios D’Meyumad or both? The halacha is to recite the brachah before, but in order to have the brachah apply to the latter tekos as well, we refrain from talking, or any other distraction, until the Tekios D’Meyumad are complete, toward the end of the Mussaf repitition.
The Shulchan Aruch states this halacha explicitly (O.C. 592, 3). The Rif asks whether someone who does speak should then recite the brachah a second time before the Tekios D’Meyumad. He says that prominent rabbis reprimanded those who spoke, but held that the blessing should not be repeated before the latter tekios.
The Ran then launches an extended inquiry, saying that the case of tefillin differs, since the transgression is to cause an additional, superfluous blessing to be recited. In the case of the shofar blowing, there is no additional brachah involved. And we do not see, continues the Ran, that once one begins a mitzvah he cannot speak until it is complete. As an example he cites Bedikas Chametz. He disagrees with the poskim who forbid speaking throughout Bedikas Chametz, saying if that were true then after reciting HaMotzi we would be forbidden from speaking throughout the meal, and after Leishev B’sukkah we would be forbidden from speaking throughout the time we do the mitzvah of eating, drinking, sleeping and relaxing in the sukkah.
The case of speaking after the first set of shofar blasts would appear to be less problematic than speaking during Bedikas Chametz since after the first set of shofar sounds we have already fulfilled the mitzvah in principle.
Despite the argument he presents, the Ran concludes that in deference to the opinion of the Reish Mesivta cited in the Gemara, one should still refrain from speaking.