Because unlike Nvidia, AMD sell CPU as well. dx's inefficiency hurt their CPU sales.
Had dx been more efficient and the FX CPUs was able to consistently push 60fps, it doesn't matter if Intel was pushing 100-120fps, it would had been overkill more most people (especially those who don't want to spend more money than they need to) would have considered them buying AMD CPUs instead at the lower price, as most of them game on 1920 res 60Hz monitor, rather than 120Hz monitor.
The dx12 may or may not have been in development before Mantle, but nobody can honestly say for certain had Mantle not came along that MS would have considering about reducing the CPU overhead and low level access.
If dx12 can reduce CPU overhead, overtaking Mantle's effectiveness to the point of AMD has no hope of competing with them in the future, then AMD would probably consider retiring Mantle...but until then, I think Mantle would be here to stay.
I honestly think AMD bringing out Mantle is a good thing, as it forces MS to put in more effort into improving the efficiency of dx. MS had monopoly on API for PC gaming for too long...even if they released dx with poor efficiency, people would have no choice but to use it, as there is no other better alternative available. So I disagree with people arguing that AMD is trying to replace dx with Mantle, but rather it's serve as dual purposes of improving gaming experience for games that support it, as well as keeping MS from slacking off on improving dx...it's a win win scenario to them. Even if Nvidia users can't use Mantle, they would benefit from the higher standard of dx that MS are forced to bring out.