How do you think the Moon orbits the Earth and the Earth orbits the sun? Of course there is gravity in space. Gravity is space, but just distorted.
In this scene, nothing would be floating around because it would be inside Earth's gravitational field, and thus slowly be pulled towards Earth (unless it was closer to the Moon). Because Jeff, Will, and the aliens are flying inside Earth's gravitational field, they, and other objects in the ship that they are flying, are not floating around as you would expect, and given the fact that Jeff is flying towards the Earth at high acceleration, then the reality is, he and Will would be able to sit pretty comfortably (by that, I mean not float off their seats).
The circled area is the earth's gravitational field. The tugs demonstrate the force of the Earth's gra-vity and the Moon's gravity, and how they pull objects towards them. (On Earth, this would demon-strate how and why we fall to the ground after jumping in the air.) The tides are irrelevant to the to-pic, so don't worry about that. The point of this picture is to visualise what I am saying, and so in other words, the alien mothership is inside this circled area, which is distorted by Earth's mass. By distorted, I mean that objects can't float around when inside this area, and this is how gravity works (in a nutshell).
This also answers your underlying question, because since Jeff and Will are accelerating towards Earth (faster than the speed of the Earth's gravitational pull, in fact) then they won't float towards the Earth either as much as you would expect them to just float around in any direction (inside the ship that they are flying, at least).
You might have seen astronauts float around in documentaries, but that is because they are quite far away from Earth. It is unlikely that they will be away from Earth's gravitational field (because they would probably have to orbit around the Earth, like when fixing a satellite or going to the Interna-tional Space Station -- ISS for short), however it is likely that they will instead be located at the edge, where gravity is weak. Near the green circumference of the Earth's gravitational field in the picture above, this area is where you experience microgravity. The entire scene you showed was clearly close to Earth, so gravity was not weak enough for Jeff and Will to float.
One of the main details that the producers missed was the fact that there is no sound in space, so you would not be able to hear the alien mothership moving, or the explosion at 0:52. This is because molecules are so disperse in space that the development of sound is impossible. On earth, sound travels to your ears by vibrating air molecules via a longitudinal wave. Through this process, the molecules must compress and then pad out for the vibration of sound to travel through them and reach your ear, which constructs the form of such a wave.
If the molecules are distributed across very large areas, like in space, then they cannot compress, and thus sound would be impossible because there is not enough molecules to compress in order to allow for sound vibrations to pass through them and into your ear.
I apologise if this is too physics related and not quite independence day related.