Why do some people wear their tzitzis strings out for people to see, and others don't?
Shulchan Aruch (Orach Chaim 8:11) states that the tallis katan should be worn over one's clothes, "so that he will always see it and remember the mitzvos."
Magen Avraham (subsec. 13) comments that kesavim ([Kabbalistic] writings) say that it should specifically be worn under one's clothes, but he says that nonetheless the actual tzitzis should be left hanging out, "and not like those who tuck them into the corners." He also adds that one may indeed keep them tucked in when walking among non-Jews (who might make fun of him, or worse), but that at least they should be showing when reciting the blessing.
Mishnah Berurah (8:26) goes further, and sharply condemns those who wear their tzitzis tucked into their pants.
It's possible, then, that the common custom of doing so arose for the reason that Magen Avraham mentions - fear of the local non-Jews - and became entrenched. But based on all of the above, it would seem that it's better to wear them sticking out.
Wearing Tzitzit in has its source in Sha'ar HaKavvanot 7b(it is also found in the language of the Tur Siman 8). Rav Ovadya Hedayya ZTz"L writes in Yaskil Avdei 5:3 and in 8:2 extended Teshuvot for wearing tzitzit tucked in. The first is al pi halakha and the second is more al pi sod. Rav Ovadiah Yosef Shlita also writes an extended Teshuva to that effect in Yehave Da'at 2:1.
Here's what I heard from Rabbi Herschel Welcher in Queens:
Having them out provides for the visual reminder to one's self.
On the other hand, there's more Kabbalistic significance to having them tucked in.
Both are valid; thus, what a person chooses to do for himself depends on what he thinks will be a better form of serving God, given his current situation in life.
And yes, some people will wrap the strings around their belt and tuck the rest in, to try and satisfy both.
I've also heard that among those who wear theirs out, it became preferable to have out only the strings, not the knots. Back when people made their own knots, it could be embarrassing for those with less coordination to have their less-than-stellar knots showing.
Rabbi Berel Wein mentions that one of his rebbeim in Chicago wore his Tzitzis in; when asked why, he told his students that they should always remember that their rebbe wore his in, and if they were ever in a situation where tzitzis-hanging-out would look wrong, they would thus wear them tucked-in, rather than taking them off altogether.
I am unsure where he says it but just a pointer, I belive that Sefardim do not wear them out because of Kabbalistic reasons and Reb Ovdiah Yosef adds as not "LEHOTZI LAZ AL HARISHONIM".That is not to call into question the Minhag of the earlier Rabbanim to wear them tucked in.(Probably based on the reasons Alex mentioned)
Chaim Burg, in his book "For God's Sake - Perspectives on Chumrot", discusses this issue and states that it was only around the late 40's that the head of the Eidat Charedit in Jerusalem took it upon himself to influence children to wear their tzitzit out. Until then, photographs of many gedolim and Jews do not show the tzitzit worn out. It would seem that the wearing of them out is a relatively late developing chumrah.
"For God's Sake", page 130: "At this point most likely someone will bring up the point that the Mishne Brura says that tzitzit should be worn outside. But Rav Mendel Zaks, son-in-law of the Chafetz Chaim (who wrote the Mishne Brura), said that his father-in-law had expressed remorse and regret for having written that. And that is why the Chafetz Chaim didn't wear his tzitzit on the outside."
I'm just going to copy my answer for the other thread:
HUGE disagreement among the Poskim. Don't want to get too deep into it, but here is the summary (there are also different Pesakim in different cases- for all the details Yalkut Yosef is recommended):
According to the Shulhan Aruch clearly (Orah Haim 8:11, 24:1) they Lechatehila should be revealed. Rav Ovadia Yosef (Yechawe Daat 2:1) holds the Tzitziot must be tucked in (see Halacha Berura ch.8, Yalkut Yosef ch. 8 and Or Lesion 2:2:2). Rav Yaakov Hilel writes (Gevurat HaAri pg. 137) that the Tzitziot must be revealed based on the Arizal (Shaar Hakawanot 7c and Olat Tamid 39a). According to the Hafes Haim they must be revealed and he is very harsh about this (Mishna Berura 8:26).