So I recently watched the first movie in the Kara no Kyoukai series, but it just started without any explanation whatsoever. For the people that have seen the whole movie series, will they explain what is going on and why stuff is happening?
Well, yes (...mostly). It wouldn't be a particularly good series if it didn't.
The big thing to note is that the movies aren't in chronological order - movie #1 is set after some of the later ones. It starts in medias res, in other words.
If you really want to find out what the chronological order is, see the answers to this question: How do I watch Kara no Kyoukai?. But I wouldn't recommend it - each movie tells you where in the timeline it falls, so you can piece it together yourself as you watch.
In the spoiler block below, I list some mysterious aspects of movie #1 that are explained later. I do not list the actual explanations, so this is only lightly spoilery.
What's the deal with Shiki's left arm? Who is the girl with Touko at the end? Why can Shiki kill ghosts? How do Shiki, Touko, and Kokutou all know each other? Why does Fujou Kirie have those supernatural abilities? What's all that magico-babble about "records"?
There are, however, some things that won't make sense even after you've seen the entire series (7 movies + epilogue). This is a mostly because Kara no Kyoukai is set in the so-called "Nasuverse", a near-legendarium written primarily by Nasu Kinoko (the author of Kara no Kyoukai). There are some details in Kara no Kyoukai that only make sense in the context of other works in the Nasuverse. That's a whole 'nother rabbit hole for you to delve into if you like what you see in Kara no Kyoukai.