How to export the results of a word search using word 2013?

Here's one way:

1. Press Ctrl+H to open the Replace dialog.
2. Select the Find tab.
3. Type your search string into the "Find what" box.
4. Click Find In and select Main Document.
5. This will select all results.
6. Click the title bar of the document to return focus to the document without deselecting the results.
7. Press Ctrl+C to copy the selected text.
8. You can then paste the results into another Word document (each result will be a separate paragraph) or an Excel sheet (each result will be a separate cell).

It's not at all clear what you want to copy, aside from whatever your search term is. For a possible solution, that outputs the sentence the found text occurs in, try:

Sub Demo()
Application.ScreenUpdating = False
Dim StrFnd As String, StrOut As String, Rng As Range
StrFnd = InputBox(Prompt:="String to Find")
If Trim(StrFnd) = "" Then Exit Sub
StrOut = "Sentences containing '" & StrFnd & "'" & vbTab & "page"
With ActiveDocument.Range
  With .Find
    .ClearFormatting
    .Replacement.ClearFormatting
    .Forward = True
    .Wrap = wdFindStop
    .Format = False
    .MatchCase = False
    .MatchWholeWord = True
    .MatchWildcards = False
    .MatchSoundsLike = False
    .MatchAllWordForms = False
    .Text = StrFnd
    .Replacement.Text = ""
    .Execute
  End With
  Do While .Find.Found
    StrOut = StrOut & vbCr & .Sentences.Last.Text & vbTab & _
      .Information(wdActiveEndAdjustedPageNumber)
    .Collapse wdCollapseEnd
    .Find.Execute
  Loop
End With
StrOut = Replace(Replace(StrOut, Chr(12), ""), vbCr & vbTab, vbTab)
Set Rng = ActiveDocument.Range
With Rng
  .Collapse wdCollapseEnd
  .Text = StrOut
  .ConvertToTable vbTab
  With .Tables(1)
    With .Rows(1)
      .HeadingFormat = True
      .Range.Font.Bold = True
    End With
  End With
End With
Application.ScreenUpdating = True
End Sub

Do be aware, though, that VBA has no idea what a grammatical sentence is. For example, consider the following:

Mr. Smith spent $1,234.56 at Dr. John's Grocery Store, to buy: 10.25kg of potatoes; 10kg of avocados; and 15.1kg of Mrs. Green's Mt. Pleasant macadamia nuts.

For you and me, that would count as one sentence; for VBA it counts as 5...