Why are the Greeks and the Trojans fighting?
Like so much of the story of the Iliad, the war between the Greeks and the Trojans has causes on two levels -- that of men and that of the gods.
"The Iliad begins with the topic of wrath. Homer bids the Muse to sing of the wrath of Achilles and all that resulted from it: the wrath of Achilles towards Agamemnon, the wrath of Menelaus against Paris, the wrath of the gods against each other and against mankind" (eNotes).
On the human level, this is a war being fought over a woman -- Helen of Troy (e.g., "the face that launched a thousand ships"). Helen is a beautiful woman, perhaps the most beautiful woman alive, and she is married to Menelaus, brother to the greatest king among the Greeks, Agamemnon.
Paris (son of the king of Troy, Priam) "steals" Helen from Menelaus at the end of a visit wherein Menelaus' household had been a hospitable host. To recover his wife and his pride and to right the wrong that has been done to him, Menelaus gains the help of his brother and the best men of Greece to take Helen back by force if necessary.
On this level, Paris has given the Greeks cause to attack Troy. On another level, the war between the Greeks and Trojans can be attributed to the gods.
When Paris is approached to settle a dispute among the goddesses Aphrodite, Hera and Athena as to which of them is the most beautiful, he chooses Aphrodite.
"Each of the women then presented Paris with a bribe. Hera’s bribe was power and a kingdom of his own; Athene’s bribe was wisdom and success in battle; Aphrodite’s bribe was love—the love of Helen of Sparta" (eNotes).
Athena, along with Hera, sides always against the Trojans throughout the Iliad and her antipathy against Troy is explained in this anecdote.
Thus the war is in part a dispute between three gods and in part a dispute between two men.
The root of the challenges between the Greeks and the Trojans involves the actions of Paris, a Trojan, and Helen, wife of the Greek Menelaus and sister in law of the aggressive Greek king Agamemnon. After dining and enjoying the hospitality of Menelaus, Paris romances and abducts his wife and some of the Greek king's money in the still of morning, with no one noticing. The conflict between desire and duty leads Paris to embrace desire in his action. This is considered to be a very serious breach of hospitality and a direct attack to the honor of Menelaus. The collision of values between upholding one's honor and avoiding conflict leads Menelaus to place his honor above all. An insulted Menelaus joins with Agamemnon, who is skilled at stoking his brother's sense of insult, in the waging of war against the Greeks. While Menelaus fights for his his honor, Agamemnon sees the opportunity to expand Greek interests into the highly prized and protected world of Troy. Using his brother's predicament, Agamemnon assembles an army to start the war of the Greeks against the Trojans.
I think because the Trojans took the greeks queen and they were not happy. Odysseus made an exceptional plan. His ships looked like they were sailing away, meanwhile someone was giving the Trojans a horse and the Greeks were inside it so when the Trojans took the horse, when the Trojans where sleeping, when the Greeks were in the horse, the Greeks got out and killed all the Trojan men