The sentence I saw it in is:
It was against this background that Mitsubishi approached KFC with a proposal to start a joint venture in Japan.
It seems to mean "in the case", but I'm not sure and I cannot find a clear definition of it.
In that sort of use, "against this background" means something like "in this context". It's sort of saying "If you consider what I've just told you, you will understand the particular significance of the fact that..."
Read as: 'with this in mind.' I can't be certain without context, but I suspect the previous sentence informs this one.