I believe it is because the "til" should be seen as the "to" in "in order to", thus it looks in danish like "he gives us the signal in order to... to begin". In English the "in order to" part is reduced to "to" and the infinitive form is remove which leads to this confusion. To sum up: "til" shows the idea of "in order to". "at" has to be written because it is Danish grammar and it can't be remove.
I'm no native though, that is just what I understood and if has always worked so far :)
I think you're correct (also not a native) because I asked my Danish friend about it. I thought it weird that it was 'to to ... do something'. She explained it that it was more like 'in order to'.