Yaskil Avdi (vol. 8 Orah Haim 2) writes that the reason the Sisit should be under the Begadim is because the Arizal writes that the Talit Katan is בחינת זעיר אנפין בהיותו בעיבור (represents a fetus in its mother's womb) and the Talit Gadol is כנגד זמן לדתו והגדלתו (represents the time of its birth and growing up) and therefore the Sisit Katan musn't be seen out at all like a baby in his mother's womb.
I'll explain as simply as possible: when something is in his mother's womb it is concealed and that's what Talit Katan represents therefore the Sisit Katan must be concealed. However, the Talit Gadol is like when it is born and it is raised therefore the Talit Gadol must be exposed.
See the responsa of the Mahari Bruna (Rabbi in Germany in the 15th century) siman 96 who seems to write that it is considered haughtiness to wear the tzitzit exposed. This is one of the reasons many Ashkenazim also do not wear their tzitzit exposed.