Theodore Roosevelt actually “entered” politics more than once. He entered political life, left it for a time, and then returned. I will discuss both of these entries.
Roosevelt’s first entry into politics came in the early 1880s. He was attending law school at Columbia in New York City, but became interested in politics and dropped out of law school. Beginning in 1882, Roosevelt was elected to the New York State Assembly three times as a Republican. His main issues were issues of reform. He was very interested in using the government to reduce the degree of corruption in the business community. He also got involved in trying to reduce governmental corruption.
In 1884, however, Roosevelt’s mother and wife died (on the same day). Devastated by the losses, he eventually left politics to go out west and try to become a rancher. While Roosevelt did not stay long in the West, he did not return to elective politics for some years. He did remain involved in politics, campaigning for presidential candidates and serving on the US Civil Service Commission. He then became president of the New York City Police Board in 1895. In these positions, too, he was a reformer mainly interested in greater efficiency and less corruption. Roosevelt’s final appointed position was as Assistant Secretary of the Navy under President McKinley. In this position, Roosevelt helped to push for the United States to go to war with Spain. When that war broke out, he resigned his post in order to serve in the military.
After the Spanish-American War, Roosevelt reentered elective politics. He was elected governor of New York in 1898. Because of his popularity as a “war hero” and because of his visibility as governor of New York, he was selected to run as McKinley’s vice president in the 1900 election. The ticket won the election and Roosevelt became president when McKinley was assassinated in 1901.
Thus, Theodore Roosevelt entered political life twice. He entered first as a reformer and then later as a war hero who was still interested in reform and in other progressive causes.
Pohnpei397's answer provides a detailed account of Roosevelt's entry into politics but doesn't describe why Roosevelt entered politics in the first place. Though Roosevelt seemed destined for politics, he actually had little interest in politics throughout his childhood. His true passion was the natural world, and composed volumes of biological taxonomy. While at Harvard, he studied both taxonomy and government, but chose to enter politics because he felt it was a more feasible career option and more likely to provide for his new wife. Roosevelt grew frustrated with the political process, and often contemplated leaving the field (his first State of the Union describes his desire to expand executive power). He had little interest in the daily work of a politician at first, but fell in love with the campaign process and giving speeches. Edmund Morris's The Rise of Roosevelt hints that Roosevelt grew to love being the center of attention and politics allowed him this opportunity.