At first, it might start with curiosity. Harry is used to his wizarding world, and the Muggle is something new. They might meet by chance, like in a park or a coffee shop. Then, as they get to know each other, they start sharing their different worlds. Harry might show the Muggle some simple magic tricks, and the Muggle might introduce Harry to Muggle hobbies.
In many fanfictions, the relationship's development is full of twists and turns. It could start with a simple attraction. But then, things get complicated. There might be other wizards who oppose their relationship because of the Muggle - wizard divide. Harry has to fight for their relationship. Meanwhile, the Muggle is also growing, learning to be brave in the face of the strange and sometimes dangerous wizarding world. Their love deepens as they support each other through all these difficulties, and they end up having a very special bond that is unique to their situation.
Often, the relationship develops gradually. There could be misunderstandings at the beginning. For example, the Muggle might not understand why Harry has to disappear sometimes for wizarding business. But as they communicate more, they start to accept each other's differences. The Muggle might learn about the wizarding world's history and culture, and Harry might become more involved in Muggle daily life. They build trust and affection over time, facing challenges together like any normal couple would, just with the added complication of magic.