Why did Homer include instances in which Odysseus received help from the gods and instances in which Odysseus helped others overcome obstacles?

The relationship between the gods and human beings is a major theme in the Iliad and the Odyssey. Since you mention Odysseus helping others, I assume you are talking about the Iliad. In that epic specifically, Homer uses the actions of the gods and of human beings to show that humans are capable of succeeding with or without the help of the gods.
In the Iliad, the gods have personal interests in the different sides of the war. Some have children on one side or another, others are the main god worshipped by a given city or culture, and many have some other personal relationship with the Trojans or the Greeks. Some gods want the Greeks to win, while some want Troy to triumph. For this reason, the gods interfere in assorted ways, helping both sides to overcome obstacles, as you say.
The gods' actions can have quite dramatic effects on the war. In book 3, Aphrodite rescues Paris from the battle at one point, saving his life. She chooses to do so because in the past, he had named her as the most beautiful of the major goddesses. This change is important because Paris's kidnapping of (or elopement with) Helen was the event that began the war, so his death might even have ended the fighting or otherwise radically transformed the situation. Other gods make vital changes as well. Later in the epic, in book 8, Zeus, king of the gods, insists that the other gods stop interfering in the war. The gods have never taken control of the war entirely, but Zeus decides that what they are doing is too much and that they need to leave the war to human beings.
At this point, Odysseus's cleverness becomes especially important. The human beings need to work on their own without help from the gods. Odysseus is one of those who has the capacity to do so. He comes up with various tricks to help the Greeks succeed. (They ultimately win the war due to his cleverness, though the episode of the Trojan Horse—the trick that finally defeated the Trojans—is not mentioned in the Iliad, because the epic ends before that happens.) If not for Odysseus, Achilles, and others on the Greek side, the Greeks might have lost once the gods stopped helping them overcome their struggles. Instead, they were able to win due to their own human capacities.
The ability of human beings to function independently of the gods in this manner is an important theme of the epic. If Homer had not first described the gods' interference, there would be no way to demonstrate how things work with and without the gods. He needs to show both gods and humans helping the different sides to overcome obstacles in order to portray the relationship between the two in a way that supports this theme.