After their engagement, before Aragorn and the host's return, Faramir knew that while Eowyn loved him and genuinely wanted to marry him, she still had a fangirlish crush on Aragorn, that wasn't distressing to him (because yeah, who wouldn't) but he knew was a source of anxiety for her, because she was worried about how she was going to face him after everything that had gone down between them.

To help Eowyn overcome her dread (and romanticise Aragorn just a little less) he showed Eowyn the crown of Gondor, which Aragorn would be crowned with.

Eowyn was suddenly a lot less worried, and sure enough, when she saw Aragorn again, with the crown on his head, she found it much easier to look him in the eye.