Preface: This subject is complicated by changing social standards. Forced marriage has been routinely practiced around the world into the contemporary era, depending on the society. (This is a major theme of Game of Thrones, as is the condition of women in pre-modern societies in general.) The Rape of the Sabine Women can provide some context.
I see three main reasons, lurid appeal, ritual, and humanism.
Mythology and folklore serve many functions. They are form of entertainment (appeal), a commemoration of religious events (ritual), and, in the case of the Classical period in Greece, it is one of the vehicles for the birth of humanism.
This rationale is easy. "Sex sells", and violence as well. You find everywhere from the very earliest myths to contemporary media. (Gilgamesh starts out with a prostitute and a fist fight. The Osiris myth involves murder, dismemberment, and necrophilia.) Rape combines the two (see: "sexual violence") and provides a titillation.
Storytellers have to support themselves and to do that they need to hold the attention of an audience. One of the central rules of fiction is high stakes = high audience engagement. The more transgressive the better. (Kill Bill and The Girl With The Dragon Tattoo are contemporary exemplars, and both can be seen as forms of Philomel, which was a female-centered revenge story involving rape.) Rape is widely used in modern entertainment as a pretext for revenge, so this phenomenon is in no way restricted to Greek mythology.
Burkert, Eliade, and Frazer have written extensively on this subject. At a very high level, there is archeological evidence of human sacrifice in ancient Europe and other ancient cultures. Human sacrifice is a central element of several of Euripides' plays, but by the time of Euripides, these sacrifices of legend have been ritualized in the form of drama. (The Festivals of Dionysus were religious festivals, and the play The Bacchae is, at its core, a commemoration of the death of the god.)
There is an original religious function to mythology. It covers everything from the creation of the universe to the acts of the gods. In the case of the Greek gods, these acts are often lurid. The behavior of the gods is commemorated in stories, rituals (re-enactments), and in the acts of humans. Cronus swallows his own children, as do Tereus and Thyestes. Just as Cronus overthrows Ouranos, Zeus overthrows Cronus.
Persephone, in the ancient context especially, is a coming-of-age story. Her abduction and forced marriage to Hades explain the cycle of life and the seasons. It's also symbolic of death and rebirth. [See: Kore & Persephone]
Persephone's is a central myth, explaining winter, and the Kore is an archetype--in some sense, her story is the story of almost all women. Abductions and forced marriage in Greek myth may be reflections of her story.
There also the factor that Zeus is a serial rapist and so it's not surprising rape becomes a staple of Greek Mythology. A key difference is that what is permissible for the gods is not necessarily permissible for man--Zeus marries his own sister. So human rapists in Greek mythology are generally villains (Tereus and Thyestes) and, unlike the gods, typically come to bad ends, even in the case of the hero little Ajax.
In his Agamemnon, Aeschylus makes the king's concubinage of Cassandra Clytaemestra's main motivation for killing her husband (jealousy and fear of being supplanted). Although Aeschylus was quite sympathetic to Agamemnon, Euripides is significantly less so, citing Agamemnon's murder of her infant from her previous marriage, and sacrifice of their daughter, Iphigenia, at Aulis, as Clytaemestra's motivations, which leads to the next element:
The humanism wiki doesn't give the full story because the undermining of divine authority begins with works like the Iliad, which has many functions, including subversive. In this great poem the dignity of the gods is undermined while the dignity of man is elevated through Achilles and Hector. It also evokes tremendous pity, for Hector in particular--the poem, which begins with the "rage of Achilles", ends with "the funeral of Hector, the breaker of horses."
Unlike the storytellers, thinkers like Plato used mythology to explicate philosophical inquiry. (In the sense of the Iliad as a subversive text, Zeus may be said to not have escaped the prophecy on Thetis in being ultimately diminished by her son, ironically in a rare case where he forgoes having sex;)
Catharsis (pity and dread) was a central element of Greek Drama. These plays rarely commemorate the travails of the gods, instead celebrating the ordeals of humans. In the Bacchae, it is Pentheus who serves as tanist for Dionysus. Aeschylus writes of the sufferings of Prometheus in service of mankind, but also writes of the trials, and vindication, of Orestes. Sophocles most famous heroine is Antigone, who dies in service of a principle. Euripides can be viewed as strongly feminist, through plays such as Hecuba, The Trojan Women, and Iphigenia at Aulis. (Humanism is rooted in the agency of humans, and feminism in the agency of women, both central elements of Greek Drama.)
An aspect of Shakespeare that makes the works such standouts is the creation of multi-dimensional female characters, whether villains or heroines. So with Euripides.
Like The Rape of Lucrece, Ovid's account of Philomela evokes tremendous pity. Empathy is a key to seeing the other as human. Andromache is play about Hector's wife, and her life as slave, after the fall of Troy. It can argued that by creating strong, believable female characters, and in generating sympathy for their travails, authors like Euripides, Ovid and Shakespeare put a spotlight on the condition and plight of women. In an indirect sense, this supports the idea of women's rights.
Political Uses
There can also be a political component, as in the myth of Lucrece, which casts her abuser, Tarquin, as a tyrant to be overthrown. This is the rationale for the establishment of the Roman Republic. Exploring the meaning of the Latin răpĭo, rooted in abduction and seizure by force, women, as prizes of war, were "booty" in both senses of the word. The abduction of Helen is the pretext for the Trojan War, but, if there was such a war, it was surely fought over the trade route connecting Asia & Europe and the Euxine & Aegean.
Notes:
Rape, in the modern conception, seems, in pre-modern times, only considered transgressive in regard to wives and daughters specifically in an extra-marital context--i.e. only a crime when involving "free women", or, women of status, as opposed to chattel, and only outside the bounds of marriage.
Modern statistics on rape can be controversial, but under-reporting is definitely an issue. Regardless of the precise numbers, sexual violence is still common, and rape is still used as an instrument of warfare. The prevalence of rape in stories and mythology may be a reflection of how often it occurs in the real world.
I tried to keep this concise in that any one of these aspects could fill a dissertation, so I apologize if everything is not fully explicated. I can provide more details in comments. I also recognize this is a potentially charged subject and have attempted to approach it as academically as possible.
