I passionately believe that's it's not just what you say that counts, it's also how you say it - that the success of your argument critically depends on your manner of presenting it.

This quote is saying that our words do not just define us, but how we say them is also important. The way we present our arguments – with care and respect – is what determines whether our arguments will be successful. In other words, we must take into consideration how we construct our words and the manner in which we communicate them in order to get our point across effectively.