Although some Jews have a custom to wear their tzitzit tucked in to their pants, the origins of that custom remain somewhat obscure. The mitzvah is to see them and keep all of the mitzvahs in mind.
Tzitzit Tucked in or Showing?
Rabbi Eli Mansour notes that twice in his presentation of the laws of tzitzit the Shulchan Aruch writes that the tallit katan should be worn over one’s clothing, so that he sees the tzitzit at all times and thereby is reminded of the mitzvahs. According to the Shulchan Aruch, even the tallit katan itself should be worn over one’s shirt (like many Chassidim do today).
The Mishna Berura sharply condemns the practice of those who tuck the tzitzit inside their pants rather than leave them exposed, saying that this not only undermines the purpose of tzitzit – “you shall see them and remember all the commandments of God” (Bamidbar 15:39) – but also denigrates the mitzvah.
Tzitzit and Kabbalah
However, various kabbalists, including the Arizal, hold that neither the tallit katan nor the tzitzit strings should be exposed. Therefore many Sephardim, who generally follow the laws and customs of the kabbalists, keep their tzitzit tucked in. According to Rabbi Bentzion Abba Shaul zt”l, Rabbi Ezra Attia zt”l remarked that any Sephardic Jew who exposes his tzitzit casts aspersions on the previous generations of Sephardic Jews.
Meanwhile, the Magen Avraham (and Shulchan Aruch Harav) holds that the allowance to conceal the tallit katan applies to the garment, but that the tzitzit should remain hanging out.
Rabbi Chaim Vital explained the Arizal’s practice of keeping both the tallit katan garment and the tzitzit concealed by saying the tallit gadol represesents chitzonius while the tallit katan represent pnimius. The previous Lubavitcher Rebbe zt”l contended that the Arizal’s custom was not meant as an instruction to others, but his own practice, adding that on the Arizal’s level, a look inward would suffice, whereas the rest of us need to keep our tzitzit out and visible.
The question gets more complicated when it comes to techelet tzitzit. About ten years ago a rav, who did not support wearing Ptil Tekhelet tzitzit, said they could be worn discreetly, i.e. at home or not showing. But it seems to me this largely compromises the proper fulfillment of the mitzvah.
Rabbi Berel Wein reportedly said that one of his rabbis in Chicago wore his tzitzit tucked in. When asked why, he told his students that they should always remember that their rebbe wore his tzitzit tucked in, and if they ever encountered a situation where tzitzit hanging out would look untoward, they would wear them tucked in, rather than taking the tallit katan off altogether.