Literary Analysis 101: If a book has a big mean animal that just won't stop attacking, you can be 100% sure that the big, mean animal is a symbol, and you can be 98% sure that it's a symbol of man's struggle against nature.
In this case, Old Slewfoot specifically shows us Jody's changing view of nature as he matures. See, in the beginning of the book, Jody sees the bear as something exciting, thrilling, and even romantic. (It sounds to us like Rawlings is referencing Sublime: the idea that nature can be thrilling and terrifying all at the same time.)
But Jody isn't too concerned about literary or philosophical ideas. He's just stoked to go on a fun field trip with his dad, even identifying with Slewfoot: "He pictured old Slewfoot, the great black outlaw bear with one toe missing, rearing up in his winter bed and tasting the soft air and smelling the moonlight, as he Jody, smelled and tasted them" (1.105). You know, to kill the bear you have to be the bear.
When he sees an actual death caused by the old bear, though, Slewfoot—and nature—become more frightening and dangerous. Listen to him describe it:
A chill ran along Jody's backbone. He could picture the shadow, big and black as a shed in motion, moving among the black-jacks and gathering in the tame and sleeping sow with one sweep of the great clawed paw. (3.38)
He's still excited to hunt him, but he can see the danger involved now. And his attitude takes one more turn, when Slewfoot attacks after the flood and the hunt turns vengeful. Now, he sees it as a duty: "Following old Slewfoot was probably a lifetime job" (26.152). And when Penny finally shoots him, his words sum up their relationship with nature perfectly. "Well, old fellow, you was a mighty mean enemy, but you got my respect" (26.221).
That's a lot to take in, so here's the short version. Jody's changing relationship with Old Slewfoot is a microcosm of how he grows up over the course of the novel's year: at first he thinks nature is awesome and fun, and then he thinks it's scary, and then he sees it as a worthy foe that he has to conquer in order to live.