There is only one way to know: follow the path they prescribe. No amount of worldly perception or inference (no amount of science, philosophy, or common sense) can prove or disprove the reality of the Vedas or the Supreme Lord and His mercy.
The senses and intellect are material, and eternal truths are spiritual. That which is material cannot recognise that which is spiritual: it can neither confirm nor deny its existence. So, there is no use seeking proof in the realm of perception and inference for that which exists outside their reach.
So, the question really is: is the prospect they present us worth pursuing? Is eternal life and well-being a greater prospect than temporary attainments in the world of impermanence? If so, then all there is do to is follow the path they prescribe. There’s nothing to prove or argue about. As soon as we take up the path, we’ll start to experience first-hand the factual existence eternal truth and the unique fulfilment found there alone.