The author's purpose is to shed light on the relationship between Lancelot and Guenevere. He achieves his goal through careful selection of details which suggest the following:
1- Acquaintance between the two characters: they both ackowledge eachother's presence using first names
2-Excitement about being reunited: a simple greeting wasn't enough and so they "go up to eachother"
3-A degree of physical closeness: this shows as they hold eachother's hand "and stay that way for a moment"
4- A conflict involving King Arthur: the passage suggests a degree of jealousy from Lancelot towards King Arthur and some guilt from Guenevere before Lancelot regarding King Arthur
5- An attempt at damage control: Guenevere gives a short answer to Lancelot and changes the topic which suggests she cares about his feelings
Through the dialogue, the author sets the mood for what comes accross as a platonic love between the two characters.