This is a good question to reflect upon carefully and to understand that there are many facets, or levels of understanding to it. One facet does not nullify the other.
Regarding the first detail of your question, "Is it just to bear children?",the answer is clearly no.
Legitimate children are also a possible product of what is called a Pilegesh (Concubinal) relationship. And that type of relationship is not marriage. It is valid and legitimate partnership according to Torah, but is on a lower level of union than the state of Marriage.
With that brief introduction presented, let's try to set up a few areas worth considering:
Marriage in the Torah as it relates to the Jewish people is called Nisu'in (נישואין) like is used in Mishnah Ketubot 2:1.
It is basically comprised of a specially defined relationship between a Jewish man and woman which makes them exclusive to each other in the sense found in Chagigah 14a and makes them responsible to and for each other like in VaYikra 5:1, and a manifestation of partnership like in Shabbat 31a, unity and with G-d's help, ultimately a source of new life like is expressed by Rabbi Akiba in Sotah 17a.
Rabbi Akiva taught: If a man and woman merit reward through a faithful marriage, the Divine Presence rests between them.
A token or sign of that relationship is a written agreement called a ketubah. This Ketubah parallels the written Torah which G-d gave to the Jewish people at Har Sinai.
This higher relationship is basically the fulfillment of the principle of, "In all your ways, know Him."
as found in Mishlei 3:6. This expression of know Him is the same term as Adam knew Chava, his wife. It has a connotation of connection, cleaving and union.
The relationship between G-d and the Jewish people is compared to that of a husband and wife like is found in Shir HaShirim and many other places.
In keeping with that thought, the answer to the last part of your question regarding two parts coming together to make a whole, can be understood.
Marriage also embodies the Torah concept of Teshuvah, return to G-d.
Judaism teaches that G-d is one and that G-d preceded the creation and is the first cause like is expressed in the Shema and in the Adon Olam prayer. And that process of creation outlined in parshat Bereshit shows that creation, which culminated in the creation of the first man and woman who were husband and wife, starts from one source (G-d), initially being revealed as oneness (the one, first light) and evolves through increasing levels of diversity to culminate in that husband and wife.
The flow of creation was increasingly away from the revelation of G-d's oneness. This is described in Kabbalistic language as Ohr Yashar, (אור ישר, Straight radiating light).
And in that condition of the diversity of creation (an absence of oneness being revealed), husband and wife bring about the supreme revelation of that oneness, what is described in Kabbalistic language as Ohr Chozer, (אור חוזר, Returning light). The two parts become literally one flesh and are a source of new life, being faithful to the system, the arrangement, which G-d set up.
And so the highest form of cleaving to G-d, of knowing Him in all ways, is through the state of husband and wife, like is expressed in Sefer Reishit Chochmah.