Decode the chmod

Javascript (ES6), 165 161 bytes

n=>[0,1,2].map(i=>(s='User:  3Group: 68Others:58None576Read48Write476Execute475and4576only'.split(/(\d+)/))[i*2]+s[n[i]*2+1].replace(/./g,c=>' '+s[c*2])).join`
`

Edit: +1 byte to fulfill the "no tab" rule

Examples

let f =
n=>[0,1,2].map(i=>(s='User:  3Group: 68Others:58None576Read48Write476Execute475and4576only'.split(/(\d+)/))[i*2]+s[n[i]*2+1].replace(/./g,c=>' '+s[c*2])).join`
`
console.log(f("666"));
console.log(f("042"));
console.log(f("644"));
console.log(f("137"));

GNU sed, 187 163 158 (157+1) bytes

Run with -r (ERE regexp). File contains no trailing newline.

s/(.)(.)/User:   \1\nGroup:  \2\nOthers: /g
s/[4-7]/Read &/g
s/[2367]/Write &/g
s/[1357]/Execute &/g
s/(\w) (\w+) [1-7]/\1 and \2/g
s/[1-7]/only/g
s/0/None/g

Jelly, 100 91 85 bytes

^{Almost certainly golfable - 91 bytes, what?!} 8 months and 6 wisdom bytes!
- 1. more string compression;
- 2. remove the post-ordinal decrement by 48 since indexing is modular;
- 3. use better tacit chaining).

-9 bytes with the kind help of @Lynn running string compressions for me

,“£ɱ~»
Ñ
ṖK,“ and”,Ṫ
LĿK
7RBUT€Uị“ØJƓ“¥Ị£“¤/¡»Ç€“¡*g»ṭ
“ṖŒhJ"ỵd¡»ḲðJ4_⁶ẋ⁸,"j€”:ż⁹Oị¢¤Y

Test it at TryItOnline

How?

,“£ɱ~» - Link 1: pair with the string "Only"

Ñ - Link 2: call next link

ṖK,“ and”,Ṫ - Link 3: insert " and" between the last two elements of x
Ṗ           - x[:-1]
 K          - join with spaces
   “ and”   - the string " and"
          Ṫ - x[-1]
  ,      ,  - pair

LĿK - Link 4: call appropriate link and add missing spaces
L   - length
 Ŀ  - call link at that index
  K - join the result with spaces

7RBUT€Uị“ØJƓ“¥Ị£“¤/¡»Ç€“¡*g»ṭ - Link 5: construct all 8 cases
7R                            - range of 7: [1,2,3,4,5,6,7]
  B                           - binary (vectorises): [[1],[1,0],[1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
   U                          - reverse (vectorises): [[1],[0,1],[1,1],[0,0,1],[1,0,1],[0,1,1],[1,1,1]]
    T€                        - indexes of truthy values for each: [[1],[2],[1,2],[3],[1,3],[2,3],[1,2,3]]
      U                       - reverse (vectorises): [[1],[2],[2,1],[3],[3, 1],[3,2],[3,2,1]]
        “ØJƓ“¥Ị£“¤/¡»         - list of strings: ["Execute","Write","Read"]
       ị                      - item at index (vectorises): [["Execute"],["Write"],["Write","Execute"],["Read"],["Read","Execute",["Read","Write"],["Read","Write","Execute"]]
                     ǀ       - call the previous link for each
                       “¡*g»  - the string "None"
                            ṭ - tack (Jelly is 1-based so the 8th item will be indexed as 0)

“ṖŒhJ"ỵd¡»ḲðJṚ⁶ẋ⁸,"j€”:ż⁹Oị¢¤Y - Main Link: parse input and make the result. e.g.: "042"
“ṖŒhJ"ỵd¡»                     - dictionary compression of "User Group Others"
          Ḳ                    - split at spaces -> ["User","Group","Others"]
           ð                   - dyadic chain separation, call that g (input as right)
            J                  - range of length of g -> [1,2,3]
             Ṛ                 - reverse -> [3,2,1]
              ⁶                - literal space
               ẋ               - repeat -> ["   ","  "," "]
                ⁸              - chain's left argument, g
                  "            - zip with:
                 ,             -   pair -> [["User","   "],["Group","  "],["Others"," "]]
                     ”:        - literal ':'
                   j€          - join for €ach -> ["User:   ","Group:  ","Others: "]
                            ¤  - nilad followed by link(s) as a nilad:
                        ⁹      - chain's right argument, the input string -> "042"
                         O     -   cast to ordinal (vectorises) -> [48, 52, 50]
                           ¢   -   call last link (5) as a nilad  -> ["Execute Only","Write Only","Write and Execute","Read Only","Read and Execute","Read and Write","Read Write and Execute","None"]
                          ị    -   index into (1-based & modular) -> ["None","Read Only","Write Only"]
                       ż       - zip together -> [["User:   ","None"],["Group:  ","Read Only"],["Others: ","Write Only"]]
                             Y - join with line feeds -> ["User:   ","None",'\n',"Group:  ","Read Only",'\n',"Others: ","Write Only"]
                               - implicit print:
                                             >>>User:   None
                                             >>>Group:  Read Only
                                             >>>Others: Write Only