A unit rate is a singular rate for/in a (specific) relationship. Multiplying the unit rate should be able to give you the rate for 2 values in said relationship. If the multiplied unit rate and value is question are equal, then the relationship is proportional.
Let's give an example:
We have two fractions: 5/1 and 18/3. The unit rate is 5/1, since it's a singular rate for this relationship. (You can tell the number of value a rate has based on its denominator. A unit rate is 1 value, so 5/1 is a unit rate.)
The other value given is 18/3. The denominator is 3, so the value of this rate is 3. If we multiply our unit rate, 5/1, by 3 (on both the numerator and denominator's sections), then we will get the value of the unit rate when it has a rate of 3 (instead of 1). If 5/1 * 3 is equivalent to 18/3, then the relationship will be proportional because both values follow the same rate. If they are not equal, then the relationship is non-proportional since the values follow different rates.
5/1 * 3 is the same thing as 5 * 3 and 1 * 3. 5 * 3 = 15. 1 * 3 = 3. So, our value is 15/3.
15/3 is not equivalent to 18/3. This means that the relationship is non-proportional since the values do not follows the same rate.
Hope this helps! :)