The Battle of the Alamo exemplified Manifest Destiny because it was a battle that was fought to spread American ways across the continent while removing other people (in this case, the Mexicans) from power. The Texans/Americans who fought at the Alamo exemplified Manifest Destiny because they were the people who were/had been involved in increasing the area of the continent that Americans controlled.
Manifest Destiny was, of course, the idea that America was destined by God to dominate the North American continent. People who believed in Manifest Destiny believed that American culture, government, and religion were all superior to those of the Indians and the Mexicans who lived in the parts of North America that the United States did not control. They believed that the US was destined to push those other cultures out and take control of the continent.
The Battle of the Alamo and the Americans who fought there exemplified Manifest Destiny because they were part of the process of spreading American power. The Americans who lived in Texas wanted to break away from the Mexican government because they did not like being ruled by Mexicans. They thought their own ways were better and did not want to have to follow people they thought were inferior to them. The battle was fought to allow Texas to break away from Mexico and to be controlled by people who had been born in the United States.
Because this battle was fought to spread American domination across the continent, it exemplifies Manifest Destiny.