Shulchan Aruch and the Ramah on Orach Chaim 6:1 bring different explanations to the ending of this Bracha (see there for different explanations for the blessing itself).
The Shulchan Aruch explains רופא כל בשר:
רופא חולי כל בשר ע"ש הנקבים שברא בו והוציא פסולת מאכלו כי אם יתעפש בבטן ימות והוצאתו הוא רפואה.
[The phrase] "The healer of the sickness of all flesh" refers to the holes that G-d created in him to take out the waste of his food, because if it spoils while inside his stomach he will die, and taking [the waste] out of him is the remedy.
For completion I will bring the explanations for the rest of the sentence, מפליא לעשות:
1.
ומפליא לעשות מפני שהאדם דומה לנוד מלא רוח ואם יעשה אדם בנוד נקב כחודה של מחט הרוח יוצא והאדם מלא נקבים ורוחו משתמרת בתוכו הרי זה פלא
"And does miraculous things" [is in the text] because man is compared to a sack full of air, and if man makes a hole in the sack as small as the tip of a needle, air will escape, and the fact that man has many holes in him and his spirit is still kept inside him, this is a miracle.
ועוד יש לפרש על שם שבורר טוב המאכל ודוחה הפסולת
And furthermore, some explain it as referring to [how the body] selects the good parts of food and pushes out the waste.
הגה ועוד יש לפרש שמפליא לעשות במה ששומר רוח האדם בקרבו וקושר דבר רוחני בדבר גשמי והכל הוא ע"י שהוא רופא כל בשר כי אז האדם בקו הבריאות ונשמתו משתמרת בקרבו (דברי עצמו)
Gloss: And Furthermore, there are those that explain that "does miraculous things" to refer to that [G-d] keeps the spirit of man in his innards and binds something spiritual to something physical, and all this is on account of that He is the healer of all flesh because then man stays on the line of health, and his soul is kept in his innards (Divrei Atzmo).
I personally resonate with the Ramah's explanation. The intricate and complex system of a person is only possible through a spiritual source, one's soul. And we thank Hashem not only for our soul but its connection with our body, which is truly wondrous (physical meeting the divine). Perhaps that is why we say it in the morning next to אלקי נשמה.