Extract rows of data based on keyword

Below I have tried to explain the steps of a macro as clearly as possible to help you understand what is happening so that you can modify it as required.
I have also re-atached your spreadsheet with the macro loaded into it.

Sub Extract_Quoter()
Const quoter_to_extract = "Andy H"

' Every command following this, and before 'End With' that starts with a . is refering to the first sheet in this workbook
With ThisWorkbook.Worksheets(1)
    .AutoFilterMode = False ' Remove any filters currently on the data
    .UsedRange.AutoFilter field:=1, Criteria1:=quoter_to_extract ' Filter columnA (field 1), using the criteria that it must match 'quoter_to_extract'
    .UsedRange.Copy Destination:=ThisWorkbook.Worksheets(3).Range("A1") 'Copy the data shown now to Worksheet 3
    .AutoFilterMode = False ' Remove the filter again
End With
End Sub