Why does my iPhone load the apple logo and then turn off?

The iOS system software / firmware on the phone probably has gone bad. The phone has most likely either wiped all system settings and user data or performed a system update itself automatically for whatever reason.

When the phone is turned off, hold the home button while plugging the lightning cable that is currently plugged into the computer to the phone.

An apple logo will pop up, keep holding the home button until you see the iTunes symbol / recovery logo. Open up iTunes on the computer if it hasn't opened up by itself. It will then prompt you that the iPhone is in recovery mode and needs to either be updated or restored.

Try updating it first to avoid accidentally wiping the data in the case of a system update. If the problem still occurs then I'm afraid a restore has to be done as a last resort which will flash the system firmware and basically do a factory reset which will wipe all data on the phone.