Large Aperture Pro lets you add that effect to your photos and make your backgrounds look out of focus. The way the app works is that you load a photo and then mark the areas of the photo that you want to blur out.
Here's an example of what a shallow depth of field looks like when taking a picture:
and the same flower taken with a less shallow depth of field:
The app gives you some basic drawing tools, like the size of the brush and how far it extrudes from where you're touching the screen (a great little feature by the way, it lets you draw on the screen without having your finger hide the actual part you're drawing).
Once you've drawn the entire background that you want to blur, just choose the aperture size and the app will create the blur effect accordingly. The larger the aperture size (the bigger the opening on the camera), the shallower the depth of field will be, and the more blurred out the background of your image will look.
The app itself needs some improvement. When you first launch it, it's not really clear what to do. You need to first click on the icon at the top left, with the arrow, in order to load a photo. I suggest you then click on "Aperture" at the bottom and move the slider all the way to the left to choose F0.95. That will create the strongest effect. Then just play around by touching different parts of your photo and see what happens. Once you get the hang of it then try it out for real on a photo that you really want to touch up.
Once you get past that, the results you can create with a bit of patience are pretty good.