How does they move slowly in air, without falling down?
One possibility is soaring using a ridge lift - typically a situation when the wind is approx. perpendicular to a mountain ridge. The air is lifted at the front side of the ridge and an eagle can soar in the lifting air stream. This can also work without the wind,
Which is a situation of thermal flying. Typically, the ground is heated by the Sun, the air layer just above the ground is heated by conduction and at some moment it forms a kind of bubble that starts to rise. This bubble is usually long, resembling a column and lasts until the warm air is depleted. The situation can repeat (this behavior is called an interval). If a ridge is oriented south, then the Sun can create a thermal wind (intervals) that enables a bird to soar.
Can we use eagle's flying technic for flights? Yes, however, man will never be that good.
soaring: https://www.youtube.com/watch?v=63qJn9HrB7E
thermal flight: https://www.youtube.com/watch?v=KXqTCM0-zXQ
Edit: Just for completness - there exists also a wave soaring, that is reachable for gliders and maybe for hangliders, probably not possible for birds and paragliders - see pictures here : http://www.ssa.org/GliderLiftSources