Our Mishna on amud beis tells us:

מַתְנִי׳ דֶּרֶךְ תַּלְמִידֵי חֲכָמִים, עַד שֶׁלֹּא הָיְתָה בִּתּוֹ יוֹצְאָה מֵאֶצְלוֹ, אוֹמֵר לָהּ: כׇּל נְדָרִים שֶׁנָּדַרְתְּ בְּתוֹךְ בֵּיתִי — הֲרֵי הֵן מוּפָרִין. וְכֵן הַבַּעַל, עַד שֶׁלֹּא תִּכָּנֵס לִרְשׁוּתוֹ, אוֹמֵר לָהּ: כׇּל נְדָרִים שֶׁנָּדַרְתְּ עַד שֶׁלֹּא תִּכָּנְסִי לִרְשׁוּתִי — הֲרֵי הֵן מוּפָרִין. שֶׁמִּשֶּׁתִּכָּנֵס לִרְשׁוּתוֹ — אֵינוֹ יָכוֹל לְהָפֵר.

The practice of Torah scholars is to ensure that a woman about to be married should not be encumbered by any vows. A father, before his daughter would leave him through marriage, would say to her: All vows that you vowed in my house are hereby nullified. And similarly, the husband, before she would enter his jurisdiction, i.e., while they were still betrothed, would say to her: All vows that you vowed before you entered my jurisdiction are hereby nullified. This was necessary because once she enters his jurisdiction he cannot nullify the vows she made before that.

The idea that this is the practice of Talmidei Chachamim requires some analysis.  Why is this not the practice of all people?  It is like saying, it is the practice of Talmidei Chachamim to repent before the wedding.  While, technically, perhaps devout people pay more attention to this, but it is a practice that is somewhat uniform and advised to all Jewish people.

Perhaps the simplest answer (and I like simple answers) is that the practice originally spontaneously arose from families of sages, and not via a particular rabbinic enactment, and then noting the worthiness of the custom, the sages of the Mishnah endorsed it. (There are other examples in Shas where the rabbis saw a custom that arose spontaneously and then endorsed it into practice, see for example Niddah 66a, “Bnos Yisrael”.)  However, this explanation does not fit as well into the text, as it sounds like it was something only followed by Talmidei Chachamim, and not even required or endorsed for others.

Tiferes Yisrael (Yachin) simply comments that Talmidei Chachamim were aware of the legal intricacy that it is possible to annul a vow without being aware of its content or existence, thus it would not occur to a non-learned person that this was even an option.  This explanation suffers from the same difficulties discussed above, why would the sages not endorse and encourage this for everyone, which perhaps they did, but why then phrase it as a custom of Talmidei Chachamim?

The Shittah explains this more, offering the idea that Talmidei Chachamim are more scrupulous and careful about nedarim. The point being, not to stress that the practice should be limited to sages, rather that the origin of the practice came from a certain sensitivity and God fearing quality. 

I would like to suggest another peshat. In general, there was an expectation that Talmidei Chachamim behave with an additional level of dignity, see for example Bava Basra (8a) where they are exempted from certain public communal labor projects, as well as Rambam (Deos 5:1). If so, perhaps the chiddush of the Mishna is, EVEN Talmidei Chachamim should engage in this practice, as one might think that it is beneath their dignity to acknowledge that they or their family members engaged in frivolous oaths (which was a sign of moral looseness, see Nedarim 22a and specifically the Mefaresh.)

There is an overarching lesson here.  A person should not enter marriage with emotional burdens, and should make a clean slate.  Whatever it takes to create a renewal and freshness will allow for a better bonding and fresh start in life.  Just as Yom Kippur night we recite Kol Nidrei, which is a technical annulment of vows, but really so much more psychologically and kabbalistically, so too on the eve of a chasunah which is also like Yom Kippur (see Bereishis Rabbah 67:13), vows and so much more ought to be released and resolved to allow for a new beginning.