In D'varim 22, the pasuk commanding placement of tzitziyos on 4-cornered garments immediately precedes the pasuk describing/commanding getting married. The juxtaposition gave rise to the minhag of simultanaizing these two acts.
(Extracting Extracting instructive meaning from the juxtaposition of two verses is universally acceptable as a method of interpretation only in D'varim. The example used by the g'mara to prove this includes one of the aforementioned p'sukim.) In this issue of Yeshivat Har Etzion's Daf Kesher, the topic is addressed. Its mention is attributed to the Sefer Hamanhig, on whom the Mishna B'rura asks exactly @jake's question in the comment below: How could it be that a mitzva from the Torah is pushed off beyond the age of bar mitzva?
Extensive justification for this practice appearsIt is suggested that although an interesting symbolic connection exists between wrapping oneself in a talis and protecting one's wife or imitating God's ways, citing the pasuk is intended to be a postfix hint and not an ab initio source.
(Also see this Satmar publication for extensive treatment of the same question: פרי תמרים)