I’d like to take this a step further than the other excellent answers. Those all address that we have Kiddushin u’Nisuin.1 I’d like to address why.
It is, in fact, is a debate between the Rosh (Rav Asher ben Yechiel, 1250/59-1327) and the Rambam (Rav Moshe ben Maimon), whether it’s a mitzvah to get married at all. The Rosh (Kesuvos 1:12) writes that we must get married as a part of the mitzvah to have children. The Rambam (Ishus 1) writes that marriage is actually a mitzvah on its own from the Torah, and there is a further prohibition for having intimacy with a woman who you have not taken as your wife via Kiddushin, which takes effect on each and every act of intimacy.2
But whether Kiddushin is a mitzvah or part of a different one, it still begs the original question of why.
The Sefer HaChinuch (552) writes several reasons for Kiddushin:
- There should be something to formalize the marriage, that the man not come upon her as he would a harlot, with no prior interaction.
- A woman who goes through this process will be permanently aware that she is married and therefore will not come to commit adultery or rebel, but will rather respect and honor her husband, as a servant to his master. Thus their life will be peaceful, and society will be established with the Will of HaShem.3
- This is why the custom is to use a ring, something which is always present and a constant reminder, even though technically a tiny perutah may be used.
To answer your question fully, however, we must address two other obligations: not to be intimate without marriage, and to divorce with a get.
In respect to the former, the Chinuch (570), quoting the Rambam (Sefer HaMitzvos, Lo Sa’aseh 355) writes that if the Torah just set forth the laws of rape and seduction, one may have thought that being intimate with one outside the framework of marriage was just a monetary matter; after all, all a rapist has to do is pay the father 50 shekalim and take care of the girl, and he gets to be her husband. If it’s a monetary matter, well, it’s is up to the two parties to do whatever they want with their money. What the prohibition of pre-marital intimacy is coming to add is that it’s not just a monetary matter, but a moral and spiritual one as well; thus, the passuk states (Vayikra 19:29) that one should not “desecrate” his daughter to make her a harlot, lest the land become “lewd” and the land become “full of immorality.”
As for divorce, the Sefer HaChinuch writes (579) that since in an ideal marriage the husband and wife should work harmoniously, there must be an out if this doesn’t work. What is the alternative, that they remain married forever? Then when the wife and husband hate each other, she will go ahead and burn his property to the ground, commit adultery before his eyes - and there’s nothing he can do about it!
Why, then, must it be done with a document? Maybe he can just tell her to leave, and that will be enough. If that were the case, though, a woman could just commit adultery and claim that her husband sent her away before she committed the act, and she would be able to get away with it! (Remember that witnesses are never required on documents on a Torah level.) Further, if the husband is upset with his wife and can yell at her to divorce her, she would be divorced immediately; if he must go through the trouble of writing a document, he will have a chance to let his anger subside and to reconcile with his wife.
In summary, then, the purpose of these laws is to indicate that marriage is more than just a monetary contract for the wife’s “services”; it’s a holy bond between man and wife. It cannot simply be broken by word, as that would result in the very situations we are trying to prevent (i.e. immorality). Yes, every now and then this will result in the terrible situation of an agunah, and we do all we can to try to prevent such a situation (see Gittin 2a-3a, regarding bringing a get from overseas - for certain things, “because of an agunah the Rabbis were lenient”). Sometimes it’s not enough. But usually it is enough, and that’s a risk we’re willing to take, for the greater good.
1 I guess this is as good a time as any to point out that Jewish marriage has two stages - Kiddushin/Eirusin (the ring, though it could be several other things - see Mishnah, Kiddushin 2a) and Nisuin (referred to here as Chuppah, the white canopy - though I’m aware it’s debated when the Nisuin actually takes place nowadays). Although in practice we do these two actions together (or at least the same night), there used to be a 12-month gap in between them.
2For further discussion of these opinions, see Teshuvos HaRivash 395, and Ran, Kiddushin 16a s.v. איכא דאמרי.
3 These aren’t my words - go blame the Chinuch for being politically incorrect.