I decided to download iOS 9.0.2 for my iPhone 5. When the download finished it was stuck on a screen I have never seen before. Here's a picture:
When I plug it into my pc, iTunes doesn't find my phone for some reason.
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That image means that your phone is in recovery mode, basically it's asking you to hook it up to itunes so it can be reflashed.
That isn't something that normally comes up as part of an iOS upgrade, so i'm lead to believe that the upgrade was probably not successful.
Try rebooting the phone first. Hold down the volume down and home buttons until the screen blanks out. Does it come back or show the same message?
If it didn't work, and you're back to the same screen, you said that the phone wasn't recognized - try rebooting your machine first, then fire up iTunes. It should pop up a box saying that it's detected a phone in recovery mode:
If that doesn't work, the next thing to do is to put the phone in DFU (device firmware upgrade) mode. It presents to the computer as a different "kind" of device and may work around whatever's preventing the phone from appearing correctly to itunes.
If even that didn't work - something else has gone quite wrong, and you should contact Apple support.