Another strange thing is that the GND and VCC pins arent connected to anything
That is an effect where the power for the chip is bleeding through from the signal lines. The chip won't work properly, if at all like that but you might see some signals on the outputs. It is not recommended you operate chips like that, you could damage them.
when I connect them to the 5V and GND pins on the Arduino, the LEDs turn off
That's because the chip is now working properly and as I said that chip only sinks (pull down) current not sourcing it.