Degree is not a function--rather it is a parameter. So you do not want Degree@, (i.e., no @) and also yo want to divide by Degree in your definition of degree as in the following:
Manipulate[
With[{degree = \[Phi]/Degree},
Graphics@Text[ToString[degree]]], {{\[Phi], 45 Degree}, 0 Degree,
90 Degree}]
In this I also localized your definition of degree using With as well as removed the unnecessary Show.