Why is seidr seen as a woman's domain in Viking culture? What made Vikings see magic as a feminine thing?

Whoa, there's a lot built into this question. First, I don't think that Norse people during the Viking Age would have considered seidr to be magic, any more than we consider science to be magic. It was a way of thinking about the world, and it was just as rational or irrational as any other. In particular, it was probably a way of thinking about how human beings were related to each other, to animals, and to the natural world. The Norse don't seem to have though of themselves as belonging to a species inviolably separate from any others. Instead, people had the ability to shift species, to take on the attributes of other species, and later Norse legends record beliefs that the natural world was actually made of once-living beings, such as giants, humans, and serpents.

This brings me to my second point. If we're right in assuming that seidr was a shamanistic way of thinking about the world that allowed people to shift between categories, then we shouldn't assume that it belonged to a particular gender. In fact, people that perform shaman rituals that are supposed to transgress boundaries between species, etc., often transgress gender boundaries as well. So sometimes we find women buried like men, such as with spears, or men buried like women, such as with bead necklaces.

Onto my third point. Many people still performed normative gender roles during the Viking Age, and the seidr way of thinking also seems to have impacted the way they performed these male/female roles. For example, women often would brew beer or mead, which could been seen as transforming one thing into another thing, and which certainly had mythical implications, so this could be an example of seidr. Or men going into battle would often wear little icons of boars or invoke bear spirits, which could also be considered a seidr practice. So seidr was really something that could apply to both genders.

That said, your impression that seidr is mostly feminine is a common impression. In part, this is because archaeological finds have turned up some very strange women's graves, which make it look like women often would perform special ritual roles in Norse cultures. But it's also because modern people—that is, people used to living among religions like Christianity, Islam, or Judaism, which are often dominated by male hierarchies—are very interested in asking questions about religions that might have been more feminine or more democratic.

So, in sum, seidr was probably actually about transgressing boundaries, which could include gender crossing, but because it was a larger way of thinking, it could also accommodate the performance of normative gender roles. Neil Price is especially emphatic that male violence was the ugly underside of seidr. Our focus on seidr as a women's domain is more about our own cultural preoccupations than with the lived experience of the peoples of the Viking Age.

Neil S. Price, The Viking Way: Religion and War in Late Iron Age Scandinavia (Uppsala: Uppsala University, 2002).
Stefanie v. Schnurbein, “Shamanism in the Old Norse Tradition: A Theory between Ideological Camps,” History of Religions 43,2 (2003): 116-138.
Martin Carver, “Agency, Intellect and the Archaeological Agenda,” in Martin Carver, Alex Sanmark, and Sarah Semple (eds.), Signals of Belief in Early England: Anglo-Saxon Paganism Revisited (Oxford: Oxbow Books, 2010), pp. 1-20.
Lotte Hedeager, Iron Age Myth and Materiality: An Archaeology of Scandinavia, AD 400-1000 (London: Routledge, 2011).

EDIT: Thanks for the gold and for the overwhelming interest in this great question!