Nowadays, it seems just common sense to write a program in a high-level programming language and let it be compiler (or interpreted) into machine code to run a computer. However, when did the precursors of computer scientists realized that what they needed was a 'programming language' with syntax, grammar, well-defined expressions and symbols, for example. Couldn't they just thought that any machine that processes information would just get mathematical formula as input? Have 19th century loom 'programmers' already used something akin to a language, in the same way they used punched cards? Was there an eureka moment at all, when people realized that computing machines were not just some really big calculator (to do the tedious math stuff for you), but that could be used for much more (like sorting lists, printing tailored letters, and so on)?
As Conifold says, Turing's proof was a decisive intellectual step, along with Church's Lambda Calculus and Goedel's proof. And as he says, real computers then arrived with assembler language, "bottom up" language design.
Fortran and Cobol were the first major steps towards high level languages. McCarthy's LISP was also an important step: he designed it as a way to express algorithms, "top down", and we reportedly surprised when one of his graduate students decided to implement it.