For the sake of simplicity, I'll assume the rocket starts off due east, and quickly climbs and accelerates into orbit. It then turns off it's engines and coasts along it's orbit. We'll ignore that first bit, and think about what happens once it's 150km above Iceland and heading due east.
The first thing to realise is that, as far as something in orbit is concerned, there is nothing special about Iceland. Or the north pole, or the equator or anything else. The only thing that makes these points special is that the earth rotates around it's axis, but once the ship is in orbit it doesn't matter how the planet rotates below it. So lets forget about the rotation of the earth for a minute. Now all points on the surface are the same, and you can launch from any one, in any direction, and you'll end up in an orbit around the planet. If you launch east from Iceland, you'll end up in this orbit (viewed edge on):
Image adapted from www.webglearth.com
If we turn the globe to look straight at Iceland, you can see how an orbit can pass through Iceland going east and still be centred on Earth's centre of mass. If we take that same orbit and draw it on a map of the Earth, it would have the familiar sinusoidal shape we associate with inclined orbits. That's just what happens when you draw a circle on a globe, then unwrap it in the (actually slightly odd) way we traditionally unwrap a sphere to make a flat map.
So we were ignoring the planet's rotation there. What happens if we put that back in? Very little changes. Because the planet is rotating, Iceland is moving eastwards fairly fast, we can take advantage of that to make the rocket launch easier. Let's assume that we launch in the same direction, achieve the same speed, but use a little less fuel doing so. The second change is that once the ship is up in orbit, the planet rotates underneath it. If you look back at the picture above, you can imagine the planet rotating about it's axis, and the orbit staying put. When we flatten out the globe to make a map, this means the sinusoidal shape seems to move across the earth.