Scout makes her comment about parting company with Jem after the pants incident.  Jem, Dill, and Scout tried to pass a note toBoo Radley that would make him come out of his house.  Boo was a neighborhood boogeyman, but Dill was obsessed with him.  He felt sorry for him and thought he would feel better if he got to know them.

Jem accidentally leaves his pants at the Radley place, and tells his father that he lost them playing strip poker.  He sneaks back at night to get them, because he is afraid that Atticus will punish him when he finds out.  Jem explains to Scout that it is not the punishment he fears; it is Atticus’s reaction.

He blew out his breath patiently. “I—it’s like this, Scout,” he muttered. “Atticus ain’t ever whipped me since I can remember. I wanta keep it that way.” (Ch. 6)

Scout tells Jem not to risk his life just so that he won’t get spanked, but to Jem it is about Atticus’s opinion of him.  He doesn’t want Atticus to know that he did something he wasn’t supposed to.  He wants his father to think of him as more grown up.

Scout realizes that she and her brother have different priorities.  She would not have cared if Atticus found out.  She was afraid to go back to the Radley place and be shot by Nathan Radley.  Jem was willing to risk it to keep Atticus in the dark.

It turns out that Boo Radley understood.  To keep Jem from getting in trouble, he sews up the pants and leaves them for Jem to find.  Jem's reaction to and understanding of Boo Radley also demonstrates how he is more mature than Scout.  She is still afraid of him.

Jem and Scout are about four years apart in age. As Jem approaches puberty, he will seek the company of boys more, doing the deeds that boys do and girls never understand. Jem's maturity level also begins to separate him from Scout as he understands what Atticus means when he warns them off of the Radley family and takes on the representation of Tom Robinson. Scout merely obeys her father because she knows he is the authority figure. Scout is too young to understand that the male/female roles of the time period will begin to separate her from her brother. Scout is relegated more and more to sitting with Miss Maudie and Calpurnia (later Aunt Alexandria). The gender roles were much more defined in those times. Now, girls and boys can be friends and share some adventures together as the gender roles are much less stringent. Scout is young enough that she can be a tomboy, but the time of skirts and lady-like interests will be coming her way within a few years. 

In this particular scene, Jem has decided to go back to the Radley House in the middle of the night to retrieve his lost pants. The three children had previously been surprised by a shadow on the Radley Porch (probably Boo himself), heard a shotgun blast above their heads, and had to explain why Jem was standing in his underwear. Scout was full of "bewilderment" at Jem's decision to return to the Radley's once again. She feared for his safety, and she waited up for him.

    There he was, returning to me.

Jem returned to his cot, trembling. It had been an exciting night for Jem, Scout and Dill.

In Chapter 6 of To Kill a Mockingbird, Scout senses that changes are taking place in Jem. After they are caught sneaking around the Radley house and Jem's pants catch on a fence after he attempts to peer into a window, and Mr. Radley's shadow appears, then fires off a shotgun.  Of course, Jem has had to climb out of his pants; when the children are caught in their mischief by Atticus, who comes out after the gun blast, Dill lies, telling the adults that they were playing strip poker.  Jem mitigates this lie by saying that they were playing with matches, instead.  At this point, Scout declares,

I admired my brother.  Matches were dangerous, but cards were fatal.

 However, later, in his maturation, Jem becomes brave enough to retrieve his pants from the fence at the Radley house where earlier he tried peeking in the window.  Before he goes, he tells Scout that he does not want to get into trouble with Atticus, and if he leaves his shorts on the fence, he certainly will.

Maybe so, but--I just wanta keep it that way, Scout.  We shouldn'a done that tonight, Scout.

It is then that Scout declares that they first began to part.  She has had moments of incomprehensibility of Jem, but this is a major moment, for she does not understand his guilt and his acknowledgement of the attitude of Atticus, that they should be respectful of the Radley's desire for privacy.

Scout realizes that her brother is looking at the world from a more adult perspective.

Atticus had been telling Scout and Jem to leave Boo Radley alone for a long time.  They had mostly ignored his orders, because they were intent on making their reclusive neighbor come out.  Their friend Dill in particular believed that if Boo just sat on the porch with them for ice cream, all would be fine.  

Things get particularly bad one night when Jem, Dill, and Scout are trying to get a look at Boo Radley and Jem ends up losing his pants.  The children explain their absence to the adults by telling them they were playing strip poker.  The problem is they still don't have the pants.

When Jem tells Scout that he is going back to get his pants so that Atticus will not know what really happened, she panics.  She is afraid that Nathan Radley will think that he is a prowler and shoot him.  No pants are worth that. 

He blew out his breath patiently. “I—it’s like this, Scout,” he muttered. “Atticus ain’t ever whipped me since I can remember. I wanta keep it that way.”

... “You mean he’s never caught you at anything.”

“Maybe so, but—I just wanta keep it that way, Scout. We shouldn’a done that tonight, Scout.” (Ch. 6)

Jem means that they should not have disobeyed Atticus.  He is starting to understand the importance of integrity.  Lying about the poker is about to catch up with him, and he can't bear for Atticus to find out the truth and be disappointed in him.

Scout doesn’t understand why it is so important to Jem to go back and get the pants.  She tells him, “It’s not like he’d never speak to you again or somethin‘…”  For Jem, it is about the loss of respect.  Scout is still looking at the world from a child’s perspective, but Jem is starting to think like an adult.  That is why she doesn’t understand him.  The difference in years between them is showing.

Jem goes back for his pants, and finds them sitting on the fence.  They have been sewn, inexpertly.  Boo Radley wanted to make sure that Jem found them and tried to prevent Jem from getting in trouble.  Jem didn’t completely understand all of that at the time, and the pants’ presence mystified and disturbed him a little.  The incident demonstrates how involved Boo Radley is in the children’s lives.  He is already a inextricable presence, very aware of everything that is going on in their lives, even before they know it.