The splitter cable you're using combines the 2 stereo inputs into the 3 pin XLR connector, putting the left channel in pin 2 and the right in pin 3, or the other way around. You can only use that type of connection with a system that specifically uses XLR type connectors for stereo signals.
That's not the case with normal performance and PA devices, where the XLR connectors are used to carry a single channel balanced signal.
In an balanced connection, the input device will subtract the inverted polarity signal from the normal polarity signal in order to get the proper signal for that channel.
So, when feeding your stereo signal to a balanced XLR input, you're subtracting your left channel from the right (or ther other way around) and that's what you're feeding to your amp. Only audio components that are different in each of the stereo channels will be audible, everything that's equally panned to both channels will disappear.
If your mixer only has (balanced) XLR inputs (otherwise, as has been pointed out in some comments, the simplest way would be to use the RCA or Jack line level inputs), what you need is to feed each channel of the stereo signl to a separate XLR input. Since you don't have a balanced signal to begin with, you will have to feed the signal to pin 2 of the XLR connector and leave pin 3 disconnected.
Since you already have one Y cable, you can do that with another of these Y cables. Find out wich RCA connector is connected to pin 2 of the XLR connector and use that one in both Y cables (just leave the other RCA connector free in both Y cables, taking care not to let its contacts touch anything, to avoid interferences). You can also get a pair of passive converters like this one and use normal RCA cables for more flexibility in physically positioning the pieces of equipment.
I'm assuming that the stereo source will be positioned near the mixer. If that's not the case then Edwin van Mierlo's recommendation of using an active device with a balanced output is applicable. A balanced connection, due to it's greater noise immunity and stronger signal in the receiving end, is highly advisable for any distance greater than 10 meters.
Anyway, connect each output XLR connector to a input channel on your mix and hardpan each of these two channels left and right as required to recover the original stereo panning.