You need a light toward the back, otherwise known as a hairlight or rimlight. This isn't a light directly behind the camera, but one coming from further back on the side.
You can see in the picture you posted that the main light is coming from the back left.
This creates the reflections on the raindrops that lets you see them better. As you can see in the photo, it also creates hilights on her hair and shoulder.
Observe the picture which you posted. First thing that comes to mind are these 2:
- Use flash
- High enough shutter speed to freeze the subject, low enough shutter speed to not completely freeze the rain.
Try out these two components, and experiment with others as well, like using various filters afterwards, bumping up the highlights, etc.