I understand Rashi's explanation there to be that the point of dispute by the showbread is whether removing and replacing the bread are considered one process or two processes. According to the Sages they are considered as two distinct actions; thus, if you don't replace the bread at the same moment as you remove it, the table will be considered as not having bread during that interval, which would be a violation of the command for it to constantly have bread. According to R. Yosi, though, the replacing is simply the conclusion of the action of removing. Thus, since it is all one process it is considered as if the table had bread constantly — even during the interval when the technical reality was that the table did not have bread.
When we apply this back to the case of circumcision, the Sages hold that the parts of the circumcision that do not invalidate it if not performed are considered as distinct actions/processes from the actual necessary components of the circumcision. Thus, if you already stopped doing the circumcision, you have no right to initiate a new violation of Shabbat for something that would not invalidate the circumcision if not performed. However, R. Yosi holds that it's all one process, and what you are doing now is simply the completion of the process of circumcision, and therefore it is still covered under the original Shabbat violation that was permitted for the circumcision in the first place:
ומה אני מקיים תמיד שלא ילין שלחן בלא לחם ולרבנן אי שקלי ושבקי ליה והדר מסדרי לא מיקרי להו תמיד אלא התחלה אחריתי ונמצא שמתעכב שלחן בלא לחם וגבי פירש נמי כי הדר אתי מילתא אחריתי היא ולרבי יוסי דאמר כי שבקי והדר מתחלי נמי תמיד הוא דחדא מילתא הוא הכא נמי פירש חוזר אע"פ שאין מעכבין דכולה גמר מילתא היא
And how do I fulfill "constant"? That the table [merely] should not sleep without bread. And according to the Sages, if you take and leave it and then go back and set it up it is not called "constant"; rather it is the start of another [action], and it turns out that the table was left with no bread. And by [the case where] he [already] completed [the circumcision] as well, when he goes back it becomes another action. But according to R. Yosi who says that when you leave it and come back and start [putting the new bread in] it is also [called] "constant" since it is one process, here too [when] he has [already] completed [the circumcision] he can go back even though [the things he is going back for] would not invalidate [the circumcision if not performed], since it is all the completion if that process.