Try
Solve[Abs[Sin[x]]==Abs[x^2-4],x,Reals]
Mathematica will often assume variables and solutions may be complex numbers. Putting that Reals into Solve tells it to only look for solutions that are real. For this problem that seems to help it understand what to look for. There is also the issue of when to use [] versus () and when to use == versus =. And to use Abs[] instead of ||, which has a completely different meaning in Mathematica.
