Why does the USB hub need to be connected with my PC to charge my Droid phone?

Note that the USB Specification somewhere says that devices that are only connected but haven't "authorized" yet may only drain 100 mA instead of the usual 500 mA. It could be that your USB Hub simply does not supply the full 500 mA if it isn't connected to a PC, to be in line with the standard.

Wall chargers ignore that as they aren't USB devices but simply Power Bricks that output 5V/500mA on a connector that looks like a USB Port.

Edit: Copy/Pasting from Wikipedia:

Some non-standard USB devices use the 5 V power supply without participating in a proper USB network which negotiates power draws with the host interface. [...] The typical example is a USB-powered [...] battery chargers (particularly for mobile telephones), [...]. In most cases, these items contain no digital circuitry, and thus are not Standard compliant USB devices at all. This can theoretically cause problems with some computers; prior to the Battery Charging Specification, the USB specification required that devices connect in a low-power mode (100 mA maximum) and state how much current they need, before switching, with the host's permission, into high-power mode.

I think the answer is simpler. The two data lines only need to be shorted to each other and not connected to any voltage. I tried this with my Droid and it works. This seems to be part of the USB specification (from http://pinouts.ru/Slots/USB_pinout.shtml):

"A simple USB charger should short the 2 data lines together. The device will then not attempt to transmit or receive data, but can draw up to 1.8A, if the supply can provide it."

I connected my Droid to a 5V power source using a USB cord that I had opened up, cut the two data lines (white and green) and shorted them to each other on the Droid side. It charged perfectly.

USB is a host-based protocol, relying on the PC for a lot of things. It's quite possible that the way the Droid is designed, it disables power flow to the connector unless it detects a host connection or the wall adapter.

I've also heard of certain cases for the Droid causing charging issues, but that's probably just for cases that have a passthrough connector between the one on the phone and the data cable.

I suppose it's also possible that the hub is not quite providing enough power without both the computer and the hub power connected, if it's defective.