You want to build the transform of the child bones by appending them to the transforms of the parent bones.

First, some abbrebiations to make things simpler:

Mb: body matrix
Ma: arm matrix

Sb: body scale
Rb: body rotation
Tb: body translation
Sa: arm scale
Ra: arm rotation
Ta: arm translation

If your body matrix is defined like this:

Mb = Sb * Rb * Tb,

then your arm matrix would be defined like this:

Ma = Sa * Ra * Ta * Mb
or
Ma = Sa * Ra * Ta * Sb * Rb * Tb

Basically everything that you put on the left hand side adds to the chain of transforms. In this case the arm translation is only the offset from the center of the body to the arm position, it has nothing to do with the arm's global position. Because you're chaining it together with the body translation, you can say that the arm translation is defined in "body space", instead of "world space", where you would supply the actual coordinate of it. Again the arm rotation is only the rotation of the arm in its own local space, it doesn't have anything to do with the rotation of the body.

You could do a similar thing with a hand mesh, by defining its rotation and translation relative to the position of the arm.

Mh = Sh * Rh * Th * Sa * Ra * Ta * Sb * Rb * Tb.

This concept comes up a lot when doing animation, basically in the simplest explanation is that when you have an object whose transform is a derivative of some parent, then you want to define the position of the sub object relative to the coordinate space of the parent, don't try to compute it directly in the world space. Its the same concept whether you're dealing with hands on arms on torsos or moons orbiting planets orbiting stars.