"Vine aquí para oír tus palabras."
Translation:I came here to hear your words.
That is correct. (And that is how I usually think of "para", in one meaning.)
Does this mean 'I came here to hear what you had to say'? I'm not expecting people to write that as a translation but 'I came here to hear your words' is a little... stilted.
If you're asking whether this is some kind of common Spanish idiom that would more naturally translate to a more common English phrase, the answer is no. This is just another way of expressing the idea, but using these particular palabras. We use this somewhat odd expression in English too. You'll hear/see "hear your words of wisdom," for example. My sister-in-law was taught to instruct her daughter to "use your words" when she was a toddler acting out.
Obviously, this is not something you run across in everyday conversations. But, I think it's easier to decipher and translate than "I came here to hear what you had to say" - vine aquí para oír lo que tenías que decir (or something along those lines).
What is the function of para here? I responded 'i came here in order to hear your words' and it was deemed incorrect. At the very least you'd think that para was 'for', but that would only again translate as 'in order'.
Yes, I agree with singe_du_nord. "I came here to hear your words" is literally correct. But I really like the more natural and meaningful translation, I came here to hear what you had to say."