The interactive at Scale City is very helpful with understanding how scale works in the real world. Here is a link to Scale City, and you can use it with triangles and rulers to prove that the two items are similar to one another with equivalent ratios when comparing their side lengths and interior angles.
The scale factor is a synonym for constant of proportionality or rate when talking about corresponding sides of figures. It is the number that multiplies something to equal the next corresponding unit.
In the image below, the arrows show the scale factor from one triangle to the other, and vice versa. The scale factor is dependent on which figure you are multiplying, as shown in the image.
Notice that the scale factors are reciprocals of one another since 1.5 = 1 1/2 as a fraction, this is the same as 3/2 which is the reciprocal of 2/3. So the scale factor changes to its reciprocal as you change which figure you are multiplying.