If you turn the power off while a picture is being recorded to the card the light will continue to blink for a few seconds. After the image is done being recorded, the power will turn off automatically. Whenever this light is blinking, do NOT open the card slot and take the card out, as this can corrupt the image being written.
Some SDXC cards might not be recognized, depending on the card reader and your computer’s OS. If this is the case, you can connect your camera to your computer using the interface cable, and then transfer the images using the EOS Utility software.