As far as I know he was doing engineering and became interested in the foundations of mathematics and went to Frege and upon his advice he went to study logic from Russell.
So what happened which caused him to change his interests? Is there any historical statement supporting his conversion?
There were not "conversion" at all, but a progressive involvment with logic and language.
For historical evidence, see e.g. Letter to B.Russell [Nov.1913], with refernce to Bedeutung [reference] (a key-term of Frege's Philosophy of language) in relation to facts and proposition.
For W's philosophy of language in the Tractatus (1921), you can see Wittgenstein's Logical Atomism.
For early philosopical interests of Wittgenstein, we can see : Notes on Logic (1913); for details, see Michael Potter, Wittgenstein's Notes on Logic, a book fully dedicated to the first (unpublished) W's work, deeply involved with Russell's and Frege's philosophies about logic and language.
I believe it is because he noticed while studying engineering that mathematics, despite its usefulness, beauty and clarity avoids state, context, and justifiably limits its expression to its own domain and as a result does not provide us with a sufficient medium in regards to communicating with the world around us in most aspects of life. The point of mathematics is then limited to providing us with a way of measuring a non state-conscious representation of our conceptual models of the world. This is not a failure of mathematicians or Mathematics as a field, simply a recognition that Mathematics is not a sufficient language for many aspects of life, nor do Mathematicians at large appear to make any such claims to the contrary. Mathematicians are the best we have at providing those aspects of that necessary tool and we need to focus on how to best utilize the tool provided.
Language is in turn our primary tool of communication. Being our primary tool of communication, it is simultaneously both the most liberating as well as the most limiting aspect of our existences. If I can't express myself, I remain trapped inside myself.
In my opinion, we should strive to develop the most clear and efficient tools for all of us to use when interacting with each other and trying to express ourselves.
To get at this question, we have to reflect on the goals of the analytical philosophy movement that Wittgenstein was drawn into. Analytic philosophy of Bertrand Russell's day was trying to find a grounding for philosophy in the empirical world, mostly by constructing a math-based system of logic that he thought mirrored the kind of logic used in the natural sciences. The movement was targeted against rationalist philosophy — the kind of deeply inner-reflective argumentation we see in European philosophers, like Hegel, Heidegger or Sartre — and with that in mind it tried hard to establish clear relationships between statements of logic or philosophy and real-world objects or contexts that the statements refer to, or denote. And so we find Wittgenstein, at the end of the Tractatus, saying:
6.53 The right method of philosophy would be this. To say nothing except what can be said, i.e. the propositions of natural science, i.e. something that has nothing to do with philosophy: and then always, when someone else wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions. This method would be unsatisfying to the other — he would not have the feeling that we were teaching him philosophy — but it would be the only strictly correct method.
[...]
7 Whereof one cannot speak, thereof one must be silent.
So at this stage in his thinking, Wittgenstein has reduced philosophical language to statements that can be expressed in the form — the language — of the natural sciences, and rejects every other kind of statement as non-philosophical.
As you can see, the entire analytical philosophy perspective was inherently linguistic from the get-go, but the intention was to provide what Popper would later call 'demarkation': a systematic way of pushing off statements that were not considered properly philosophical, i.e., those that are irreducible to mathematical logic. However, this approach produced a number of now-famous paradoxes: Russell's paradox, the morning-star/evening-star paradox, etc. These paradoxes pointed out a weakness in the analytical philosophy approach, in that it assumed (and in fact insisted on) clear one-to-one denotation. Ambiguous denotation — referents in statements that had unclear, abstract, or variable objects — create problems for the underlying mathematical models that the movement wanted to invoke.
Russell and most of his cohort kept trying to dig into math and logic to shore up or resolve this difficulty — you can see that in Wittgenstein's quote above, which comes from his earlier work — but Wittgenstein eventually embraced this not as a problem of math, but as a problem of language itself. He decided to reconstruct the act of denoting as a functional convention of human interaction, not an objective feature of the world, and that drew him straight down the rabbit hole into deep language theory.
