The only changes i made were in the second For Each loop where you were displaying the images.
All i did was added a <table> tag (which you might need to remove. you may have your page set up differently.
I also pulled the <tr> tag to the outside of the loops. If you left them in the loop then on each loop it was creating a new row.
Then i added this if statement.
If x Mod 5 Then
'do nothing, we just need to display the image.
Which basically says if x Mod 5 then Dispaly the image,
Dim oFS, oFolder, oFile
Set oFS = Server.CreateObject("Scripting.FileSystemObject")
'Set Folder Object To Proper File Directory
Set oFolder = oFS.getFolder(FILE_FOLDER)
intCounter = 0
ReDim Preserve FileArray(oFolder.Files.Count, 1)
For Each oFile in oFolder.Files
strFileName = oFile.Name
FileArray(intCounter, 0) = strFileName
FileArray(intCounter, 1) = "<a href=" & strFileName & "><img src=" & strFileName & "></a>"
intCounter = (intCounter + 1)
intRows = uBound(FileArray, 1)
intCols = uBound(FileArray, 2)
For x = 0 To intRows -1
For z = 0 To intCols
If z > 0 Then
If x MOD 5 Then
EchoB("<B>" & oFolder.Files.Count & " Files Available</B>")