That's pretty much it.
The Father asked all his daughters what they wanted him to get while he attempted the journey to reclaim their wealth; the Daughter (who we are likely playing) only asked that he returned safely rather than asking him for a material object like her sisters. But the Father cared deeply for her as well and wanted to get her at least something. He came across a hidden mansion and stayed there for shelter; someone even seemed to be giving him food and preparing him a place to rest, as though knowing they had a guest. The next day, the Father decided to explore the grounds some more and came across a beautiful garden. He saw the rose and picked it for her, feeling it was the perfect gift.
What he hadn't expected was for the mansion to belong to the Beast; he was furious that the Father seemed to take advantage of his hospitality by taking something from his garden and demanded compensation. The Father begged for his life, as he still needed to return to his children, one of them being the Daughter who he picked the rose for. Hearing this, the Beast agreed to let him live (and in some stories, even offered to give him something that will help the man return to his wealth once more). But there was one condition: The Father was to give the Beast the Daughter who he intended to give the rose to (or just one of his daughters in general). The Father begged him to ask for anything else, but the Beast refused.
When he returned, he told his daughters about the deal. Hearing this, the Daughter who he wanted to give the rose to chose to be the one to go, in order to protect her family from the possible consequence of refusing the deal. At least, that's how some versions of the story went.