In the simplest of terms, John faints because he is overcome with what he sees. His wife has subsumed herself within the wallpaper and the designs she has perceived within it. As a result, John comes in the room and finds a very intense sight around him:
"What is the matter?" he cried. "For God's sake, what are you doing!"
I kept on creeping just the same, but I looked at him over my shoulder.
"I've got out at last," said I, "in spite of you and Jane. And I've pulled off most of the paper, so you can't put me back!"
Now why should that man have fainted? But he did, and right across my path by the wall, so that I had to creep over him every time!
The fact that he is described as "crying" out when he speaks to his wife, represents how he is unable to fully fathom what he sees. As a result, his fainting is about the only semi- appropriate response he can generate. Simply put, the sight of the wallpaper torn off and his wife circling around the room as if she is in the wallpaper is too much for him to take. He faints as a direct result of what he sees around him.
The narrator "creeps" because of her perception that she has found a way out of the wallpaper. Her creative energies had been thwarted with the "rest therapy" prescribed to her. The narrator had wanted to write, and find some outlet to express her feelings. However, under "strict" orders she had to "rest." As a result, her mind became fixated on the wallpaper. The design of the wallpaper and what she could see in it began to occupy her thoughts: "For outside you have to creep on the ground, and everything is green instead of yellow. But here I can creep smoothly on the floor, and my shoulder just fits in that long smooch around the wall, so I cannot lose my way." That she does not want to get lost in the pattern of the wallpaper is why she "creeps" around the floor. She "creeps" in order to stay in line with the wallpaper, and so that she does not become lost in this world. She has not committed suicide. At one point, she suggests that she cannot do that because it would be "improper and might be misconstrued." Rather, she finds a sense of connection to "creeping" around the room and becomes subsumed with the condition offered in the wallpaper.
Whether the narrator has escaped the established medical community's approaches is a matter of concern. On one hand, I think that a case can be made that the narrator has gone insane. To argue this, it becomes clear that she has become subsumed by the wallpaper and has lost her sense of rational thought as a result. However, I think that another good case can be made that she has challenged the establishment's diagnoses. The narrator did everything asked of her, but the results were disastrous. She has exposed the failure of the patriarchal system, a condition that her husband was a part of. This might be why he faints; a feeble response to his own complicity in making his wife the way she is. At the same time, the narrator has demonstrated how wrong the diagnosis of "simply needing to rest" actually was. There was something more complex and intricate going on that John and her brother missed. She has "escaped" in so far as she has proven the establishment to be futile in understanding her condition."In spite of" men like her husband and brother, the narrator has challenged this institution. The narrator has demonstrated an effective response to their seemingly transcendental statement of absolutist understanding. It is in this light where she has "escaped" in a manner of speaking.