How does change in aircraft's weight equal fuel mass flow rate multiplied by 'g'?
In the SI system the weight is measured in Newtons [N = kg*m/s^2], while the mass is measured in kilograms [kg].
To convert kilograms in Newtons, i.e. a mass measurement in a weight, you multiply by the acceleration of gravity g.
In this case you're converting a change in mass to a change in weight, the conversion is the same, but both sides have an additional time component in the denominator due to the derivative.
Given your comment, it seems you're confused by the good old mass-vs-weight thing. Let's have Wikipedia:

In common usage, the mass of an object is often referred to as its weight, though these are in fact different concepts and quantities. In scientific contexts, mass is the amount of &quot;matter&quot; in an object (though &quot;matter&quot; may be difficult to define), whereas weight is the force exerted on an object by gravity. In other words, an object with a mass of 1.0 kilogram weighs approximately 9.81 newtons on the surface of the Earth, which is its mass multiplied by the gravitational field strength. The object's weight is less on Mars, where gravity is weaker, and more on Saturn, and very small in space when far from any significant source of gravity, but it always has the same mass.