The horse's name, Teabiscuit, is a play on that of Seabiscuit, a real-life thoroughbred racehorse, ungainly and mistreated in his early life, who was rehabilitated by an empathetic trainer and, by the time of this cartoon, had gone on to become a racing champion and a sentimental favorite of the American public. The cartoon's story is an affectionate screwball parody and celebration of Seabiscuit's rise to fame.
At the time this short was released, sporting events were a major in person entertainment event, though the leagues known today (NFL, NBA) were still in their formative years, as was the television broadcast. Horse racing, boxing, wrestling, and more were all followed by fans nationwide, who saw film result recaps at the movies on a newsreel. And the show also included animated features like this one, which often had a current event slant (Seabiscuit, here).
The animators accurately captured the ground-up view of the horses racing overhead. It was, and still is, a popular camera angle in productions of all kinds.
The title is a parody of Seabiscuit (1933-1947), a champion Thoroughbred racehorse in the United States.