I will identify what is wrong in your code and then show you what it should be. 
First: don't use select, see This SO Question 
Range("B1").Select
Second: using End(xlDown) searches down until the last used cell before the first blank cell, if there is a blank cell before the last filled cell, you will not select the whole used range.
Range(Selection, Selection.End(xlDown)).Select
Third: setting your range variable to UsedRange, selects every used cell, negating the range in Column B; e.g. If your used range was Range("A1:F20"), then Rng = Range("A1:F20") 
Set Rng = ActiveSheet.UsedRange
Forth: you start your loop from the last cell in Rng to the first cell in Rng; e.g. using the above example, your loop would start at Range("F20") and the next cell in the loop would be Range("E20"), etc. 
For i = Rng.Cells.Count To 1 Step -1
Fifth: Your If statement checks for the text in each cell of Rng. Thus, it will check Range("F20") first, and since your text is not in a cell until Column B. So, the first and subsequent cells will be True for <> and the row will be deleted. The loop will still check every cell left in the row and delete the row again-and-again, and then start the next row with the same outcome. etc.    
If Rng.Item(i).Value <> "M of E" Then 
    Rng.Item(i).EntireRow.Delete
End If
Below is basic code to loop through each cell in column B, and delete the rows that don't have "M of E", 
Sub DeleteRows()
' DeleteRows Macro
'you should always set the workbook and worksheet
Dim ws As Worksheet: Set ws = ThisWorkbook.Sheets("Sheet1")
'assign a last row variable using .End(xlUp).Row
Dim lRow As Long: lRow = ws.Range("B" & ws.Rows.Count).End(xlUp).Row
    'loop from the last row to the second row(skip header)
    For i = lRow To 2 Step -1
        'use Cells to set the row and column and check for "M of E".
        If ws.Cells(i, "B").Value <> "M of E" Then
            'If the cell in Column B doesn't contain "M of E", delete the row.
            ws.Cells(i, "B").EntireRow.Delete
        End If
    Next i 'loop
End Sub