In this novel, Jess feels like he has to be the fastest runner for two main reasons: to make his father proud and to make a name for himself at school.
At the beginning of the story, we learn that Jess has been running all summer, in preparation for his ascension to fifth grade when school commences. Jess is resolved to be the best runner in fifth grade. To him, it's the only way he can get his father to take notice of him again:
Maybe Dad would be so proud he'd forget all about how tired he was from the long drive back and forth to Washington and the digging and hauling all day. He would get right down on the floor and wrestle, the way they used to. Old Dad would be surprised at how strong he'd
gotten in the last couple of years.
The above quote highlights Jess's loneliness; he has four sisters and only gets along with one, May Belle. Meanwhile, his two older sisters, Ellie and Brenda, keep him at a distance. In fact, they are often dismissive of him. As for his mother, Jess knows that she loves him; however, her love doesn't make up for his father's emotional neglect. So, Jess resolves to be the fastest runner in order to turn his father's attention to him again.
The second reason Jess wants to be the fastest runner is to earn some notoriety for himself among the lower grades: this would be the third, fourth, and fifth grades. In the story, we learn that the lower grades tend to be sidelined at the field during recess (the higher grades usually claim the best athletic equipment and play areas). So, the third, fourth, and fifth graders have resorted to informal running competitions among themselves.
Jess wants to be the best because he rather likes the idea of the third-grade boys following him around "like a country-music star." He's tired of being known as that "crazy little kid that draws all the time." There doesn't seem to be any glory in such a label. As for being the best runner, Jess has only won one race, and that was during fourth grade. For one whole day (and until lunch the next day), he was known as the fastest runner among the third-, fourth-, and fifth-graders. That brief period of notoriety fed his desire to dominate the running field completely.