Do I believe
A question with many layers. Like asking, do I believe in prayer and ritual. What is implied in your question is: do I believe witchcraft achieves the end results it is supposed to. Well, that implies I know what those end results are indeed supposed to be. Again, implied in your question is the assumption of a tangible end, a change, a transformation... But without clarifying what kind of change or transformation I'm being asked to "believe in" or evaluate, it is a difficult question to answer. You see, I have seen from my experience that Wiccan practitioners have many different end goals, some which may be intangible, such as having a lovingkind heart.
Like others here have said, witchcraft is more than the Disney version of magic; where you get everything you want just by waving your wand. That is the capitalist version of magic, implying that something can come from nothing, that no effort or offering must be given in order to receive. That's not what I've seen of actual practitioners of Wicca, at all.
Committed practitioners endeavor for spiritual enlightenment, connection with the Earth, communion with the Earth, respect for All, and, yes, a scientific (but non invasive) study of nature, the natural laws governing nature, and a serious questioning of what our moral ethical and spiritual obligations and roles are as human beings, given our ultimately completely interconnected nature with the Earth, and other beings.
Ultimately Wicca is about Love, and coming to love. All the rituals and physical tools Wiccan use are supposed to remind us to enter back into the reality of love, interconnection with all beings, and harmony and balance within ourselves and others.
Having experienced the reciprocal creation of such a spirit of love, I believe it is possible; and waving a wand is only 1/100th of the creation of such a spiritual abiding space. However, I respect the traditions and innovations that Wiccans choose to physically symbolize this spirit of love, harmony and balance. These rituals and symbols help us to learn and understand again the reality of interconnectivity we live in, not just on an intellectual level, but an emotional and spiritual awareness which we can bring to our daily living.
A long answer to a seemingly simplistic question. This answer should also be understood as only one of many possible truths or understandings; it is rooted in my personal experience and I acknowledge it as such. Other experiences, having been learned from in depth communion with lore, ritual, and inquiry should also be validated as personally legitimate.
We are our stories and this is only one story.