Make an inversedict

Mathematica, 28 bytes

Merge[Reverse/@Normal@#,#&]&

The input is an Association, the output is an Association too.

Explanation

Normal@#

Convert the input Association to a List of Rules.

Reverse/@ ...

Reverse the Rules.

Merge[ ... ,#&]

Merge all Rules, grouping duplicates with a List and then with the identity operation. Creates an Association.

Usage

Merge[Reverse/@Normal@#,#&]&[
 <|"Clyde" -> "blue", "Sarah" -> "blue", "Fred" -> "green"|>
]

<|"blue" -> {"Clyde", "Sarah"}, "green" -> {"Fred"}|>

Python 2, 54 bytes

lambda i:{i[k]:[a for a in i if i[a]==i[k]]for k in i}

Usage:

f=lambda i:{i[k]:[a for a in i if i[a]==i[k]]for k in i}
print f({"Clyde": "blue", "Sarah": "blue", "Fred": "green"})

Gives

{'blue': ['Sarah', 'Clyde'], 'green': ['Fred']}

Brachylog, 10 bytes

tᵍ⟨ttʰhᵐ⟩ᵐ

Try it online!

Takes input as a list of [key, value] pairs through the input variable, and outputs a list of [value, [keys]] pairs through the output variable.

 ᵍ            Group the elements of
              the input
t             by their last elements.
         ᵐ    For each group,
   t          its last element
  ⟨     ⟩     paired with
      hᵐ      the first element of every element
     ʰ        with the first element of the pair being replaced by
    t         its last element
         ᵐ    is the corresponding element of
              the output.

...such an eloquent explanation...