There are at least two ways of learning about influence lines. The first way is algebraic, as given in the solution quoted. If you have learned about shear force diagrams before coming to influence lines, it is easy to get the two things confused. It is necessary to keep returning to the definition of an influence line to remind yourself that it is NOT a shear force diagram. Part of your question implies you may have taken on this confusion.
The second way of learning about influence lines is to sketch the deflected shape of the beam when a unit displcement is given at the point of interest. In this case, C is the point of interest. So giving the beam a unit displacement in shear at C (0.5 down, and 0.5 up in this case) gives the deflected shape of the beam as shown in the solution. You could (and should) use this approach to check a solution obtained algebraically. This approach also works for other functions such as bending moment (but that will do for the moment!)
