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.

Why does Jess want to be the fastest in his class in Bridge to Terabithia?

Jess needs to be the fastest in his class because he likes winning.  The narrator tells readers that Lark Creek Elementary does not have a lot of athletic equipment.  Anything that the school does have, the sixth and seventh graders nab right away.  The third through fifth graders are left with nothing other than having foot races across the field.  The previous school year, during one of those races, Jess had beaten everybody else, and he got a real taste for winning.  

One time last year Jesse had won. Not just the first heat but the whole shebang. Only once. But it had put into his mouth a taste for winning.

Jess needs to be the fastest kid in the class, because it will change the way other kids look at him.  The narrator also tells readers that Jess is the odd kid who draws a lot.  

Ever since he'd been in first grade he'd been that "crazy little kid that draws all the time." But one day - April the twenty-second, a drizzly Monday, it had been - he ran ahead of them all, the red mud slooching up through the holes in the bottom of his sneakers.

For the rest of that day, and until after lunch on the next, he had been "the fastest kid in the third, fourth, and fifth grades," and he only a fourth grader.

Jess believes that if he can consistently be the fastest kid in school, his classmates will revere him instead of thinking that he's odd.  

He could hear the third-grade boys screaming him on. They would follow him around like a country-music star. 

Lastly, Jess needs to be the fastest kid in order to earn his father's approval.  Jess desperately longs for his dad's love and attention.  His dad works long hours, so he is not home very much.  The dad also isn't fond of Jess doing art.  Jess figures that if he excels at running, his dad will be proud enough to want to spend more time with him.  

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. 

In Bridge to Terabithia, why does Jess want to be the fastest fifth grader?

Jess wants to be the fastest runner for several reasons, and readers should look to chapter one for textual evidence. I think one reason is a simple old-fashioned competitive drive. The desire to be the biggest, strongest, fastest, etc. is something that competitive kids and athletes share. Jess has this drive. Being one step down on the podium is not something he strives for.

He had to be the fastest—not one of the fastest or next to the fastest, but the fastest. The very best.

Second, Jess is good at running, and he knows it. Running gives him a chance to show that he is athletically talented at something. Plus, Jess likes the feeling of winning. Nobody likes losing, but some people aren't too bothered by it. Some people are like that about winning too. Winning is nice, but it doesn't make or break their day. Jess isn't like that. Winning really makes him feel good.

One time last year Jesse had won. Not just the first heat but the whole shebang. Only once. But it had put into his mouth a taste for winning.

Finally, and most importantly, Jess wants to win to impress his father. Jess feels that his dad is disappointed in Jess's desire to draw and do art related stuff. By being the fastest runner, Jess can prove to his dad that he has athletic talent.

Even his dad would be proud. Jess rounded the corner. He couldn't keep going quite so fast, but he continued running for a while—it would build him up. May Belle would tell Daddy, so it wouldn't look as though he, Jess, was a bragger.

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.

Jess prepares himself to meet his goal like any other athlete. He practices, and he practices hard even if it means waking up earlier than normal.

His body was begging him to quit, but Jess pushed it on. He had to let that puny chest of his know who was boss.