To understand what these gyroscopes and accelerometers are measuring, you first need to know the physics and geometry behind "Degrees of Freedom". In terms of physics, each degree of freedom (DoF) in a motion that can exist in 3-dimmensional space. For example, 1DoF is moving along a straight line, or "axis"
Each time you increase the DoF, you exponentially increase the number of possible positions. A 2DoF system is usually (but not always) two axes that are perpendicular to one another and in the same plane. Picture a typical X-Y plane. A 3DoF system is similar, but it adds the 'z' axis as well, allowing for movement inside of a field, instead of just on a plane.
Once you get beyond 3DoF systems, to four through six, you usually start talking about rotation around each of the previous three axes. Picture a free-spinning wheel, that can move up and down the axle (guess the engineer forgot the collars and bearings). So if you have a 3DoF system where an object can also rotate about each axis, you really have six degrees of freedom. This can be visually demonstrated by the "Right Hand Rule"
Take you right hand in front of you and make a fist. Now give a thumbs-up. Keeping the thumb up, point your index finger out. Finally, keeping your thumb up and index finger out, point you middle finger parallel to your chest, perpendicular to both your index finger and thumb. Imagine that each of these fingers is an axis, with your thumb being "Y", middle finger being "X", and index finger being "Z". Move your hand up, and you're moving along the Y-axis. Move away from you, and it is the Z-axis. To the left and right is the X-axis. This is a full 3DoF system. But you can rotate your hand as well, and 'around' each finger too. This gives you three more degrees of freedom, so it is a full 6DoF system.
Usually, when you start talking about a system that is greater than 6DoF, you are almost always talking about multi-jointed system. I'll use a robotic arm as an example. In robotics, each time you add one joint to an arm, you increase the degrees of freedom of the system by one. As you get higher and higher degrees of freedom, the math gets exponentially more complicated and the system gets harder to control as a result. This is part of the reason why we don't have bionic limbs just yet.
But in this case, when these chips talk about "9DoF", they really mean they can measure acceleration, instead of just change in position. They do this by monitoring inertia and doing some calculus to figure out how quickly and how far it just moved. Because the movements they are measuring are pretty small, they are probably only measuring linear motion along its X-Y-Z axes (instead of rotation around any axis).
The 6DoF chips measure motion along 3 perpendicular linear axes, as well as rotation around of each of these axes. The 9DoF chips do all this, plus measure linear acceleration along each of these axes.