What does the "On Error Resume Next" statement do?

It basically tells the program when you encounter an error just continue at the next line.

It's worth noting that even when On Error Resume Next is in effect, the Err object is still populated when an error occurs, so you can still do C-style error handling.

On Error Resume Next

DangerousOperationThatCouldCauseErrors

If Err Then
    WScript.StdErr.WriteLine "error " & Err.Number
    WScript.Quit 1
End If

On Error GoTo 0

When an error occurs, the execution will continue on the next line without interrupting the script.

It means, when an error happens on the line, it is telling vbscript to continue execution without aborting the script. Sometimes, the On Error follows the Goto label to alter the flow of execution, something like this in a Sub code block, now you know why and how the usage of GOTO can result in spaghetti code:

Sub MySubRoutine()
   On Error Goto ErrorHandler

   REM VB code...

   REM More VB Code...

Exit_MySubRoutine:

   REM Disable the Error Handler!

   On Error Goto 0

   REM Leave....
   Exit Sub

ErrorHandler:

   REM Do something about the Error

   Goto Exit_MySubRoutine
End Sub