Once you've finished writing your name, also write the date.
Literal
Name [object-を] write-finish [conditional-たら], date [also-も] write-keep.
書き終わる is a compound verb ('write' + 'finish') = 'finish writing.' たら gives a sequential 'once X happens, then Y.' 書いておく uses ~ておく ('do in advance/for future use') to mean 'write [it down] for later reference.' A staged instruction for filling out a form.