The concept of tosefet first arises regarding Yom Kippur. The gemara[1] learns from the Torah’s mention of fasting from the 9th of Tishrei[2] in the evening until the next evening that the fast begins a little before the end of the 9th of Tishrei and ends a little after the end of the next day. The gemara demonstrates that one should similarly refrain from melacha[3] directly before and after both Shabbat and Yom Tov. (The concept is applied to Shemitta years as well.[4]) The Rambam omits this concept of tosefet in the laws of Shabbat, and even in the laws of Yom Kippur, he mentions it only regarding fasting.[5] However, the Shulchan Aruch[6] requires tosefet Shabbat.
There are a two or three possible elements of the period of tosefet: 1) One violates tosefet if he does melacha up until Shabbat begins.[7] 2) One’s acceptance of the new day is binding even if he accepted it earlier than necessary.[8] 3) It is a mitzva or even an obligation to accept tosefet actively. Your question addresses #3, whose source is unclear.
As a source for #3, some cite the Mishna Berura,[9] who says that accepting Shabbat is accomplished orally. He in turn cites the Rama,[10] who writes that cognitive acceptance without speech is ineffective, and one who only mentally decided not to eat any more before a fast may therefore eat again. The Mishna Berura[11] also cites important poskim who rule that a mental decision is binding, and he does not seem to decide between the approaches. Based on these passages, the Shemirat Shabbat K’Hilchata[12] says that one accepts Shabbat before the day’s end, preferably with speech, although his acceptance is possibly binding with a cognitive decision alone.
There are two troubling factors here. First, few observant Jews or shuls are careful to actively accept Shabbat early. In fact, many shuls finish Mincha moments before or even after sunset, at which time tosefet has kicked in automatically.[13] (It is unclear how long tosefet is, but it is almost certainly single digits of minutes.[14]) It is also strange that the Mishna Berura does not cite a source for the requirement to perform an act of accepting Shabbat early. The Rama he cites refers only to element #2. The Rama writes that oral voluntary acceptance is binding. It is possible that it simply makes sense to say that if there must be tosefet and speech creates it, one should actively accept it rather than be forced into it (although we note that tosefet is not accepted actively before Shemitta or at the end of Shabbat). However, it is still troubling that the classical halachic sources do not mention this requirement, even regarding Yom Kippur, the original source.[15]
In fact, it is not clear that explicit acceptance is necessary. Rav Ovadia Yosef[16] argues that if one is in a shul where Mincha (which must be prayed before Shabbat) will finish after sunset, he may daven without accepting Shabbat. Rav Yosef claims that the essence of tosefet is simply refraining from melacha. The Ohr L’Tzion[17] also posits that the requirement of tosefet is to refrain from melacha and understands the Mishna Berura to be referring only to element #2.[18] He does add, however, that there is a value to an oral acceptance, if one is possible. The Shevet HaLevi[19] is also not convinced that active acceptance is necessary. While he notes that many Rishonim consider tosefet a mitzva, he says it is likely that mental acceptance is enough.
There is little reason not to actively accept Shabbat a few minutes before sunset when one is unlikely to need to do melacha. However, the fact that most people do not do so is not a mistake. For Ashkenazi women, the matter is a non-issue, as they usually accept Shabbat when lighting Shabbat candles.[20] When they do not do so, their status is like that of a man.