Anna and Elsa created a (regular, non-magical, inanimate) snowman when they were children, and named it Olaf. However, after Elsa accidentally hurts Anna using her powers, she tries to hold back her powers and avoid Anna. Later, during a time of extreme stress, Elsa's powers got out of control, freezing Arrendele. Elsa sings the song "Let It Go", which has themes of previously trying to bottle things up, and how she's now letting go. During that song, she creates a snowman with the same appearance as the one she made with Anna as a child, before she withdrew from her relationship with Anna. The implication is she is letting herself revisit her early childhood with Anna. So the animate Olaf is based on the one she built years ago, but is not the "same" one, and did not survive through the intervening summers.
Anna recognizes the similarity, and takes it as a symbol of their previous closeness, and how it's possible to regain that closeness. Elsa creating a snowman that was clearly inspired by the one they made together shows that there's a part of her that longs for that closeness, and Anna appeals to that.