It was cold at dawn, so I put on an extra blanket.
Literal
Dawn [at-に] cold-was [because-ので], blanket [object-を] 1-more-sheet pulled-over.
The counter 枚 counts flat objects (paper, blankets, clothing, tickets, plates). もう１枚 = 'one more sheet.' 毛布を掛ける uses 掛ける ('to hang, put over') in the sense of pulling a blanket over oneself — one of the many senses of this polysemous verb.