The dual HDMI outputs on the BDP-10xEU series of players provide additional versatility in the connections you are able to make to the rest of your equipment.
The player's Dual HDMI Output setting defaults to Split A/V, which allows video output to a display and audio output to an AV receiver / processor, but this setting can also be changed to Dual Display in order to accommodate connecting to two displays.

Using Dual Display mode will result in both audio and video being sent to each device that is connected via HDMI, but both the audio and video quality may be limited compared to using a single HDMI connection or Split A/V mode.

The player's Output Resolution cannot be adjusted per HDMI output, so it will be limited to the highest resolution supported by both devices. For instance, if one display is capable of receiving a 1080p signal, but the other connected display cannot receive a signal higher than 720p, the Output Resolution will be limited to 720p for both outputs.

Audio is handled similarly in Dual Display mode, though it is often less of an issue since many displays do not contain any Dolby or DTS decoding capabilities. As such, audio will often be limited to 2ch PCM when using Dual Display mode.