Let's start at the beginning so you'll understand this better. There's a piece of data missing with the numbers you provided.

First, you need to understand the concept of "gear inches". Gear inches (also called development), is how far the bike will travel with each revolution of the pedals. The distance varies based on the following formula:

gear inches = wheel diameter * teeth in front chainring / teeth in rear sprocket

So, given your 20t setup, you would have 20 inches * 42t/20t = 42 gear inches

On edit:
This means that for each revolution of the pedals (in those gears), the bike will move forward 42 inches times pi (use the circumference of the wheel (pi * d)).

Now, changing the 20t sprocket to 16t, you would have 20 inches * 42t/16t = 52.5 gear inches

This means that changing from the 20t sprocket to the 16t sprocket, for each revolution of the pedals you would travel 10.5 inches times pi further.

When you add the element of speed to the discussion, you have to know your cadence which is the rate at which your are spinning the pedals. Given your app's statement that you would be traveling at 15.2 mph (it has to know your cadence to provide this information), changing from the 20t to the 16t sprocket would yield a speed of 15.2 * 52.5 gear inches / 42 gear inches. This would yield 19 mph. This means that making the switch, if you maintain the same cadence, would increase your speed from 15.2 to 19 mph.

You cannot convert gear ratios to speed without considering cadence. Without extra effort, you would not see any increase in speed at all. Going from 20t to 16t, without extra effort, will increase your distance per pedal revolution (i.e. gear inches) but it would correspondingly decrease your cadence because it would be harder to pedal.

By the way, there are more things to consider when changing a single sprocket. You should also consider the jumps between the next higher and lower gears. There's something to be said for keeping a consistent relationship from one gear to the next.

Hope this helps.