This is actually a great question!
The plain answer is YOU DON't KNOW what they say is correct but you can EXPERIENCE its Truth!
But let me ask you this:
How do you know that what you read in your physics book is correct?
For example, all physics books say that there are different phenomenon such as E = MC2 isn't it? Have you yourself physically converted the mass into its energy and conducted experiments? Then how do you believe that it is valid?
The answer is you trust in the physics book that these text books contain actual results of various scientists who performed their investigation and you accept it.
Similar with scriptures Hindus place their trust in the ancient wisdom of the sages who were able to write down with great detail and accuracy that it is indeed valid.
How do you know that you love your parents?
For example, everybody says that they love their parents isn't it? But how do you know that you love your parents? Have you seen that Love? Have you touched it? Have you smelt it? Have you tasted it? Have you even heard of it? NO. But still you know without even 2% doubt that everyone loves their parents (there are exceptions ofcourse) THIS IS BECAUSE you have experienced it yourself! Thus you know without a doubt that it is indeed true.
Similar it is with the Hindu scriptures. The Truth that they speak of, cannot (generally) be seen, hear, touched, smelt etc. However it can be EXPERIENCED! The advanced Hindus and other Bhaktas have directly experienced the Truth (partially or completely) and thus they know without a doubt that it is indeed True.
Thus what is the solid proof for Hindu scripture?
Like any other book, the solid proof of scripture lies in:
- Trusting in the wisdom of the sages and then carrying out experiments to experience the Truth.
- Experiencing theTruth directly, thereby dispelling all doubts, All the best Sir!
But HOW does one carry out these experiments? How does one experience the Reality?
How will I verify? How to be happy? How to realize God? What is the practical way?
According to this path, one does self-introspection. One asks oneself questions such as: "Who really am I?", "How long will money stay with me?", "Wherein does lie true happiness? Is it external to me? Or is it internal?", "What is the cause of suffering?", "What is it that if I will understand, I can be always free and happy at all times"
Such knowledge that arises liberates one from bondage giving one the taste of supreme bliss. [Bhagavad Gita 7:16, 17, 18, 19]
According to this path, one performs actions but without desire for the rewards of such service. By performing such actions, one gets liberated from all sufferings and obtains a direct experience of the Ultimate Bliss. [Bhagavad Gita 3:5-17]
According to this path, one loves God, with heart, mind and soul (sounds familiar?) and is constantly engaged in worship of the Lord. By praying and serving God always, one obtains the direct experience of Supreme Bliss. [Bhagavad Gita 6:40-47]
NOTE: some would say that one path is inferior and another path is superior to the other. Let us not get into such arguments unnecessarily. Obviously one is free to believe that His path is the best of all paths. But let us not deride others and their paths.
Thus basically crux to the whole answer is: indeed the scriptures cannot be verified by using senses, but they can be verified by following the various paths mentioned above. All the best. By following these paths one will be able to understand the reality of scriptures. One will obtain ultimate bliss.