As this question has generated a great deal of literature, I shall confine my answer to the most pertinent points.
This allegation is made by Orientalists, just as it was by their predecessors: Christian and Jewish writers who deeply resented the spread of Islam. The first people to make it were the Prophet's own opponents, as we read in the Qur'an:
[When Our clear revelations are recited to them, the Unbelievers say, of the Truth when it comes to them: "This is evident sorcery!" Or do they say, "He has forged it"?] (Al-Ahqaf 46:7–8)
They were desperate to protect their interests against the rising tide of Islam and hoped, as do their modern counterparts, to spread doubt about the Qur'an's divine authorship so that Muslims would start doubting its authority as well.
The Qur'an is unique among scriptures in two respects, which even its detractors accept. First, the Qur'an exists in Arabic, its original language and one that is still widely spoken today. Second, its text is entirely reliable. It has not been altered, edited, or tampered with since it was revealed.
In contrast, Christianity's Gospels have not survived in their original language; the language of the earliest surviving version of these scriptures is a dead language. In addition, their texts have been shown to be the work of many people over generations, edited and re-edited, altered and interpolated, to promote sectarian interpretations. They have lost their authority as scriptures, and serve primarily as a national or a cultural mythology for groups whose remote ancestors created their particular versions. This is, more or less, the Western scholarly consensus on the status of these once-divine books.
For almost 200 years, Western scholars have subjected the Qur'an to the same rigorous scrutiny. However, they have failed to prove that it was subjected to a similar process. They discovered that Muslims, like Christians, sometimes split into disputing factions. But unlike Christians, all Muslim factions sought to justify their position by referring to the same Qur'an. Other versions of the Gospels might be discovered or uncovered. However, all Muslims know only one Qur'an, perfectly preserved in its original words since the Prophet's death, when revelation ended.
Muslims also have a record of the Prophet's teaching in the Sunnah, the record of how he implemented Islam in daily life. Many, but certainly not all, of the Prophet's actions and exact words are preserved in the Hadith literature. These two sources could not be more dissimilar in quality of expression or content. All Arabs who heard the Prophet speak, regardless of religious affiliation, found his words to be concise, forceful, and persuasive, but nevertheless like their own normal usage. When they heard the Qur'an, however, they were overwhelmed by feelings of rapture, ecstasy, and awe. One senses in the Hadith the presence of an individual addressing other people, a man pondering weighty questions who, when he speaks, does so with an appropriate gravity and in profound awe of the divine will. The Qur'an, on the other hand, is perceived immediately as imperative and sublime, having a transcendent, all-compelling majesty of style and content. It defies sense and reason to suppose that Qur'an and Hadith have the same origin.
The Qur'an is absolutely different from any human product in the transcendence of its perspective and viewpoint. Occasionally in a few scattered phrases or passages of other scriptures, readers or listeners may feel that they are in the presence of the divine message addressed to humanity. In the Qur'an, every syllable carries this impression of sublime intensity belonging to a message from One Who is All-Knowing and All-Merciful.
Furthermore, the Qur'an cannot be contemplated at a distance, or discussed and debated in the abstract. It requires us to understand, act, and amend our lifestyles. It also enables us to do so, for it can touch us in the very depths of our being. It addresses us in our full reality as spiritually and physically competent beings, as creatures of the All-Merciful. It is not addressed to just one human faculty, such as philosophical reasoning, poetic or artistic sensibility, our ability to alter and manage our environment, or political and legal affairs, our need for mutual compassion and forgiveness, or our spiritual craving for knowledge and consolation. The Qur'an also is directed to everyone, regardless of age, gender, race, location, or time.
This transcendence and fullness can be felt in every matter that the Qur'an mentions specifically. For example, caring for one's elderly parents is placed beside belief in God's oneness, and providing decently for a divorced wife with reminders to be conscious of the All-Knowing and All-Seeing. While the reasoning behind such placement is God's alone, His believing servants know and can report its effect: It enables the inner self-reform that makes the steady, cheerful, and humble performance of virtuous actions possible. Thus, the one who does the deed does it gracefully, and its recipient is not oppressed or humiliated by it.
The Qur'an challenges its detractors to compose a chapter that can equal it. No one has successfully met this challenge. In fact, such an achievement is impossible, for only God can assume the Qur'an's all-transcendent and all-compassionate perspective. Our thoughts and aspirations are affected and conditioned by surrounding circumstances. That is why, sooner or later, all human works fail or fade away into obsolescence, and why they are too general to have any real influence or too specific to do much good beyond the specific area they address. Whatever we produce is of limited value for just these reasons. This is stated in the Qur'an:
[Say: if all of humanity and the jinn were to gather together to produce the like of this Qur'an, they could not produce the like of it, even if they backed each other with help and support.] (Al-Israa'17:88)
The Qur'an is the word of the All-Knowing and All-Seeing, Who knows everything about His creation. It therefore comprehends and tests its audiences as it teaches. For believers, the consciousness of being before the divine message can make their skins shiver, in the words of the Qur'an, so suddenly and fully does the atmosphere around and within them change.