There are three ways you can save a route.
A button at the bottom
If you scroll down to the bottom of the results, you will find a ‘Save Route’ button.
Tapping on the info box
If you tap on the info box at the top of the results, it will bring up a menu with an option to save the route.
Dragging open a Save button
And from the Routes tab, if you drag from left to right on a recent route, it will reveal a Save button.
The delete button
Conversely, if you drag from right to left on a recent route, it will reveal a Delete button to remove that route from your recent routes.