Sita seeing deer and Ram going to kill the deer:
Once Ravana went to meet his maternal uncle Mareech who was living an ascetic life in the forest. He told Mareech that he should take the form of a golden deer to lure Rama away from the cottage so that he could kidnap Sita. Mareech tried to convince him that Ram and Lakshman were not ordinary humans but Ravana did not listen and threatened to kill Mareech if he did not obey him and help him in his plan. Mareech felt helpless and took the form of a golden deer in front of Panchavati ashram of Ram. He thought to himself that he would better die from the hands of Lord rather than a demon.
When Sita saw the golden deer, she asked Rama to capture it for her. On her insistence, Rama went hunting the deer. Before doing that he made Lakshman promise him that he would not leave Sita alone in any circumstance.