OpenGL translation before and after a rotation

Now, my question is for the elbow. Why is there a translation forward on the x-axis both before and after the rotate?

If you want to imagine how the matrix operations change the model, then you need to "read" the operations in the reverse order. This is, because the current matrix of the matrix stack is multiplied by the matrix which is specified by the new operation and the matrices are stored in column-major order (fixed function pipeline).

Start with the elbow cube

glutWireCube(1.0f);

Scale the elbow

glPushMatrix();
glScalef(2.0f, 0.4f, 1.0f);
glutWireCube(1.0f);
glPopMatrix();

Move it to the right

glTranslatef(1.0f, 0.0f, 0.0f);

Rotate the elbow

glRotatef(45.0f, 0.0f, 0.0f, 1.0f);

Move the rotated elbow to the right

glTranslatef(1.0f, 0.0f, 0.0f);

Draw the shoulder cube

glutWireCube(1.0f);

Scale the shoulder

glPushMatrix();
glScalef(2.0f, 0.4f, 1.0f);
glutWireCube(1.0f);
glPopMatrix();

Move the arm (elbow and shoulder) to the right

glTranslatef(1.0f, 0.0f, 0.0f);

Rotate the arm

glRotatef(-15.0f, 0.0f, 0.0f, 1.0f);

Move the arm to its final position (to the left)

glTranslatef(-1.0f, 0.0f, 0.0f);