How does string interpolation work in Mathematica

I think the key to understand what happens under the hood is that cells contain boxes which will be rendered by the frontend and sent to the kernel for evaluation. Using "Convert To" -> "StandardForm" will modify this box structure ("Cell" -> "Show Expression" or Ctrl-Shift-E to see the box structure before and after using it). Using convert to standardform on StringForm is an edge case which produces something that is probably causing more trouble than benefit, namely an InterpretationBox. This will show as one thing but evaluate as something else. You can create interesting effects with them, e.g. a string within an identification process (evaluate the cell that the following CellPrint creates):

CellPrint[
  Cell[BoxData[
    InterpretationBox[
       "\"I am a string but would rather be a plot!\"", 
       Plot[x, {x, 0, 1}]
    ]
 ], "Input"]
]

I probably should add that InterpretationBox of course has many use cases where it causes more benefit than trouble and that it is nothing new but has existed at least since version 4.

On version 7 I get this using Show Cell Expression (Ctrl+Shift+E) on the output cell:

If you use InputForm you get:

StringForm["x=``", 1 + 2] // InputForm

StringForm["x=``", 3]

Which makes it clear this is a wrapper.

On your version it appears that StringForm has attribute HoldRest so that 1 + 2 is preserved in that wrapper.

As far as I know no evaluation ever takes place automatically inside a string itself, therefore entering "x = 1 + 2" by itself with any kind of trim will not result in "x = 3".

As mentioned before StringForm together with ToString will give the desired results.

Here are some examples, that reproduce the functionality that is provided in other languages by sprintf (or similar functions):

myStrings = {
   ToString[StringForm["This is a floating point: `1`\n" , 
     ToString[NumberForm[5.6 10^9, NumberFormat -> (SequenceForm[#1, "e", #3] &)]]
   ]],

   ToString[StringForm["This is a integer in blocks : `1`\n" ,
     ToString[NumberForm[10^9, DigitBlock -> 3]]
   ]],

   ToString[StringForm["This is a number with funny padding : `1` " , 
      ToString[NumberForm[6.888, {7, 3}, NumberPadding -> {"_", "0"}]]
   ]],

   ToString[StringForm["This is just a string : `1` " , "FOO"]]
   };
myCompleteString = StringJoin[myStrings]

FullForm[myCompleteString]

Export["text.txt", myCompleteString, "Text"]

this will give you:

This is a floating point: 5.6e9
This is a integer in blocks : 1,000,000,000
This is a number with funny padding : ____6.888 This isjust a string : FOO 

"This is a floating point: 5.6e9\nThis is a integer in blocks : 1,000,000,000\nThis is a number with funny padding : ____6.888 This is just a string : FOO "

and a file text.txt with the expected output.

See also these discussions:

*) Converting a large floating-point number to a single-line string

*) sprintf() or close equivalent, or re-implementation?

As @Szabolcs says, StringForm does not produce a string. However, if you replace StringForm in your example with TemplateApply, that will produce the string you want. (Version 10.0 introduced this command in 2014, which postdates your question and his answer.)