If $G$ is linear (which would be the case, for instance, if it is centerless) then it is special case of the more general fact that any finitely generated subgroup of $GL_n(F)$ for a field $F$ of characteristic zero is virtually torsion-free.
So all you need to know is that the lattices are finitely generated. This, for cocompact lattices, is easy, and for instance follows from what is usually called Milnor-Schwarz lemma, which is a very general lemma about cocompact isometric actions of groups on spaces. You can find a version of it in Pierre de la Harpe's book.
For general lattices, in higher rank, this follows from property T and in rank 1 by some more geometric methods.
If not, this may fail. See this for instance: