Learning almost anything sufficiently new will do that to you. Dabble in calligraphy and letters will never quite look the same again. Get into dance or poetry and you'll appreciate entirely new nuances to music.
There is some element of truth to what they said, though. Not to take away from these other examples, but it is more than seeing new nuances, as in, it can become an entirely new paradigm of thinking. In English, things feel very limited and strict, and your expressiveness (so I find, at least) has an upper limit. With something like Latin, you can express more, with less, and in completely new ways that you would not think about at all in English.
There is more to this argument, but this is a good start, I think.
Right, but thinking is just one aspect of our experience of the world.
Skills, in a broader sense expand how and what you experience. Language does this to how we experience communication and expression. Reading in general does this too, but in different ways.
Like how you can't hear a word you know without understanding its meaning, expanded beyond language.