Go to Matching Brace in Visual Studio

Place the cursor next to an opening or closing brace and punch Ctrl + Shift + P to find the matching brace. If Eclipse can't find one you'll get a "No matching bracket found" message.

edit: as mentioned by Romaintaz below, you can also get Eclipse to auto-select all of the code between two curly braces simply by double-clicking to the immediate right of a opening brace.

In Visual Studio 2019

Ctrl + D

In Visual Studio 2017 (v15.6 and after)

Ctrl + D

In Visual Studio 2017 (pre v15.6)

(edit) This feature is now built-in in VS2017: Ctrl + E, V duplicates a line if nothing is selected, or duplicates selection. You can assign it to a different key combination, or find it in the menu:

See this reference for more information.

Pre VS2017, built-in method using clipboard

As @cand mentioned, you can just do Ctrl + C ; Ctrl + V.

Ctrl + C will copy the line if nothing is selected.

Macro solution (pre VS2017)

If you'd like to implement a more complete solution, perhaps to create a simpler keyboard shortcut or you don't want to effect the clipboard, see this guide:

Visual Basic:

Imports System
Imports EnvDTE
Imports EnvDTE80
Imports System.Diagnostics
 
Public Module DuplicateLastLineModule
    Sub DuplicateLine()
        Dim line As String
        DTE.ActiveDocument.Selection.StartOfLine(0)
        DTE.ActiveDocument.Selection.EndOfLine(True)
        line = DTE.ActiveDocument.Selection.Text
        DTE.ActiveDocument.Selection.EndOfLine()
        DTE.ActiveDocument.Selection.NewLine()
        DTE.ActiveDocument.Selection.StartOfLine(0)
        DTE.ActiveDocument.Selection.Text = line
    End Sub
End Module

To create the macro, just go to the macro explorer ("Tools->Macros->Macro Explorer" or Alt+F8) and copy paste the code in a new module. Now just assign a keyboard shortcut to it:

1. go to Tools->Options...
2. under Environment, click Keyboard
3. in the "Show Commands Containing" textbox, enter "duplicate" (this according to the name you gave the module.)
4. you should now see the macro in the list below
5. choose "Text Editor" from the "Use new shortcut in" list
6. set focus in the "Press shortcut keys" textbox and hit the combination on the keyboard you wish to use for it (Ctrl+Shift+D in my case)
7. hit the "Assign" button
8. you should now see the shortcut in the "Shortcuts for selected command" textbox
9. hit the OK button

And that's it. Enjoy!