With much thanks to /u/Pytheastic (go upvote!), I'd like to write an answer more focused on this specific question. :)

The simple answer is that Joan had the support of the king, but that's pretty much running a shell company on my part. If I had to sum things up, I would say "religion and prophecy," but that also is not very interesting in and of itself.

Joan as Holy Woman

From Joan's own testimony at her trial, it is easily apparent that she was deeply immersed in the religious culture of her time. The saints most important to her are the most popular ones, she's right with the new trend in angels, she's sold on the rising importance of devotion to the Holy Name of Jesus.

This is important because Joan fits firmly in the context of the early 15th century as a holy woman and prophet. Her visions and auditions anchor her in a tradition going back to the mid-12th century of women who used the message that they spoke and acted based on direct revelation from God.

In Joan's time, some people are starting to question the validity of holy women's claims. The initial questions themselves, though in some ways the culmination of a longer trend, are highly political as a result of the Avignon papacy and (especially) the Great Schism (ca1378-1415). In other words, they are very much tied to ecclesiastical politics.

On the ground level, what we find is much more ongoing confidence in women's revelations. 14th century saints and visionaries Catherine of Siena and especially Birgitta of Sweden are all the rage. People even start attributing to Birgitta texts that she didn't write; she's that famous and popular even among the literate classes. Birgitta (and Pseudo-Birgitta) becomes especially well-known for two things that transcend the literacy barrier: prophecy and a set of prayers.

Not everyone, but a whole lot of people, took Joan absolutely seriously as conveying divine messages directly. In very particular, Charles VII was raised in an environment where his parents firmly believed in the prophetic powers of holy women. Charles VI had given audience to Jeanne-Marie de Maillé; and Isabeau, to Marie Robine (a peasant, by the way).

And this was, of course, the key issue at her initial and nullification (rehabilitation) trials: were Joan's fake or real; demonic or divine...according to the political beliefs of the judge or witness. For a demonstration, turn to no less a contemporary authority than French "theologian &c" Jean Gerson (uh...he was Really Important; roll with it), who is infamously on the record as opposing the legitimacy of holy women...but wrote dramatically in support of Joan.

Joan and Wonders

Kelly DeVries, who is basically the authority on Joan as a soldier and commander, stresses the importance of religion in the accounts of Joan's contemporary supporters as well as her own (Joan of Arc: A Military Leader, but especially "A Woman as a Leader of Men" in Fresh Verdicts on Joan of Arc, which is, well, about this question's precise topic). He's right, although his account is based on Joan's full career, including her victories. Which, again, is a liiiittle bit of a cop-out.

I want to go back to the 15th century mindset again, to look at the overall supernatural cosmology of the era. Well into the early modern era, there's no real divide between what we would call "religion" and "magic." (Indeed, "religion" doesn't even have our meaning until the 15th century.) As with belief in revelations from God, people live in a world of wonders and miracles and saints and supernatural creatures. But as seen with growing concern with witchcraft and questions about holy women's sincerity, the boundaries are just starting to be sketched out by some people.

This is especially apparent in Joan's case. The wonders associated with her don't really have a division in what she relates about other people's support of her. They do have a divide in the mindset of her interrogators--and, because Joan is frakking awesome, she knows exactly what they're doing and keeps pace. (Seriously, read Dan Hobbins' translation The Trial of Joan of Arc. She's great.)

A big one is Joan's knowledge of and then the discovery of "her sword" in a church dedicated to one of France's most important saints. The finding of a blessed object has major precursors in the Middle Ages, especially associated with the Crusades. In the 15th century, that was more important than ever. The physical reality of objects was critical to how people saw the world and religion in a way it wasn't earlier. Second, the cult of relics and saints was, you guessed it, critical in a way it hadn't been earlier. (Think of Mark Twain's remark about there being more shards of the Holy Cross in the world--in the 19th century still!--than there could have been in the actual cross.)

According to Joan herself, people also told stories about a prophecy they associated with her and a tree/forest near her home in Domremy. But in her own words, what people said to her about this was just linked to her performing wonders. This probably included a miraculous power to heal, which was also heavily tied to holy women/living saints in late Middle Ages. (There are stories about men, women, and children all trying to touch Joan, which seems suspiciously, I don't know, biblical. And yes, at a time when there was much more preaching of the Bible directly.)

The tree was associated with fairies and local children performed May Crowning-type playing/ritual activities around it, although Joan insists she never believed any of it and never engaged in a lot of the behaviors her judges asked her about. Of course, they lie WILDLY when they write up the articles of condemnation. On one hand, they say Joan admitted to various things when she categorically had not. On the other, though, they exaggerate the various behaviors and beliefs they had asked her about earlier. And, unsurprisingly, they exaggerate according to particular patterns that align with the question of fake or real, demonic or divine.

So people associated Joan with the general performance of miracles and wonders.

Hans Böhm

Okay, obviously a man, obviously German, and not obviously a few decades after Joan. However, Böhm is a crucial parallel for a few reasons. Even venturing further into the very slowly increasing fear of witches, Böhm--a shepherd from Baden-Württemberg--essentially launched an entire revolt against unjust conditions based on his own prophecies and visions of Mary.

People were ready to heed prophecy that called to them--and did.

Conclusion

Joan was awesome; she promoted her awesomeness; she did so in a way that grew out of the religious culture in which she lived and believed.

This reply by /u/sunagainstgold is a start while we wait for an answer.