What are the connotations of "Lo Sosifu ... , ve'Lo Sigre'u mimenu"?
Rashi and Ramban #1 (citing the Sifri): 'For example, not to write five Parshiyos in one's Tefilin, not to take five species with (including) the Lulav or to tie five Tzitziyos on (the five corners of) one's garment'; 1 and likewise "ve'Lo Sigre'u"
Ramban: Or even to add some words to Birchas Kohanim - such "Hashem Elokei Avoseichem Yosef aleichem Elef Pe'amim".
Ramban: See Rosh Hashanah, 28b.
Ramban: Like Yerav'am ben N'vat - See Melachim 1, 12:33. See Megilah, 14a and Yerushalmi, Megilah, 1:7. See Ramban, DH 'u'Lefi Da'ati' who concludes that we learn the prohibition against a Navi adding Mitzvos from "Eileh ha'Mitzvos" (Vayikra, 27:34), and that the Takanos that the Chachamim instituted - such as Sheniyos la'Arayos, are actually min ha'Torah which we learn from the Pasuk in Vayikra, 18:30, "u'Shemartem es Mishmarti").
What is wrong with adding to Mitzvos?
Da'as Zekenim, Hadar Zekenim, Rosh: One who adds, detracts. If you put five Tzitziyos, you detract from the four, since you did not do the Mitzvah like its law. The same applies to five Parshiyos in one's Tefilin, or five species with the Lulav. However, if you sat in the Sukah for eight days, the eighth does not come with the seven (to invalidate them, since at the time he did the Mitzvah properly). What he did is done.
Because people will be allowed to add Mitzvos, they will assume that, by the same token, why should one not be allowed to subtract.
Why does the Torah repeat below (13:1) "Lo Sosef Alav v'Lo Tigra Mimenu"?
Refer to 13:1:1:1.