Flutter ships with their default launcher icon. It is nice but you probably want a custom one. These are the steps I took to change mine:

1.- Get a the image you want to use as a launch icon as png.
2.- Save it in assets/whatever_the_name_of_the_file.png.
3.- Install flutter_launcher_icons library (link here). We are setting the icon for Android only, so add the following to pubspec.yaml:

flutter_icons:
  image_path: "assets/whatever_the_name_of_the_file.png"
  android: true

4.- Run:

flutter pub get
flutter pub run flutter_launcher_icons

The second command will generate the images in the appropriate android folders. Please note that the app cannot be running in development, otherwise you will get errors.

5.- That's it. There are other attributes that you can use for other platforms and other settings, but I didn't need to use them. Read about them in the package documentation.