Best illustrated by example. Suppose a car's fuel consumption is 24 miles per gallon. Let the number of gallons be g and the number of miles m. The relation between m and g can be written m=24g, and this can be plotted on a graph as a sloping line. The slope is 24 and it's also the unit rate: how many miles per gallon.
Take another example: a car travels at 50mph. The distance travelled in so many hours can be expressed d=50h, where d is the distance in miles and h is the number of hours. This also can be plotted and the slope is the unit rate (mile per hour) which is also what we would call the speed, rate of change of distance.
The cost of fuel is, say, $5 a gallon, so c=5g where c is the cost in $ and g the number of gallons. Again, a straight line plot has a slope which is the unit rate of cost of a gallon. The slope is 5.
We can take this further by working out another unit rate: the unit cost, cost in dollars per mile of travelling in a car that has a fuel consumption of 24 mpg and the cost of fuel is 5$ per gallon. We use the two equations c=5g and m=24g. We want m and c to be related with m as the independent variable. First, g=m/24 so c=5g=5m/24 and this can be plotted, with a slope of 5/24. So 5/24 stands for the unit rate of dollars per mile. This can be turned round: m=24c/5, where the slope of 24/5 means the unit rate of miles per dollar; in other words, how many miles can be travelled on a dollar's worth of fuel.
Just a few examples of the relationship between slope and unit rate.