Why do we get married according to Judaism? What is the point behind us working so hard to get married? Presumably it is not just to bear children! I have heard of a concept of two halves coming together to make a whole or something like that- what does this mean?
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.
There's a very simple and logical answer to that according to Rambam's view (see Ishus 1):
We are commanded to be fruitful and multiply
The Sages forbade any sexual connection outside a legal marriage, and gave it a status of prostitution.
Therefore the only way of fulfilling the Mitzvah #1 is through marrying a woman and subsequently having kids from her.
Therefore some Poskim (see Shu"A Even Haezer 1 that starts with:
חייב כל אדם לישא אשה כדי לפרות ולרבות
Every man is obligated to marry a woman in order to be fruitful, and to multiply
don't count the Kiddushin as a Mitzvah on its own, but only as preparation or instrument to Pru uRvu (#1).
Second part - Rashi on Gen 2.24 ("Hence a man leaves his father and mother and clings to his wife, so that they become one flesh."):
לבשר אחד. הַוָּלָד נוֹצָר עַל יְדֵי שְׁנֵיהֶם וְשָׁם נַעֲשֶׁה בְשָׂרָם אֶחָד
ONE FLESH — Both parents are united in the child.
THe Halachic purpose of the marriage in Judaism is to sanction the legal relations between spouses, namely in two aspects:
The financial relations between the spouses (some detailed in the Ketubah) Rambam Ishut:
- the financial obligations of the husband toward the wife
- the obligations of the wife toward the husband (not mentioned in the Ketubah)
The financial obligation of the father to support his kids from that woman. Rambam end of Ishut.
Permitted and forbidden sexual relations: (Rambam Isurei Biah)
- what women are allowed and forbidden to the husband and
- that all men are forbidden to the wife.
- the obligation of the husband to maintain regular and timely physical relations
- the obligation of the wife to allow those relations
The marriage (the social approval of the relations) is the base for all those Halachic interactions. Without marriage, there are no obligating relations between the two.