There is no effort on the part of the writer to prove the existence of God, nor the idea that He speaks and has spoken. He is apparently writing to those who accept these as truth. God exists and speaks to us.
Speech is important. We are speaking beings that live in time. Our words are so important that more often than not they are all that remains of us when we die. Millenia after their death, we the living still read the words of Socrates, Plato, Aristotle, Homer, Moses, and Jesus. The ability to speak and understand speech is hard-wired into our brains, and one of the largest areas of the motor cortex in our brains controls the organs of speech. Speech, or its proxy the written word, is undeniably our chief means of exchanging ideas of any kind.
If we posit, for the sake of argument, that there is a God that both created us and remains interested in us, would it not be our expectation that He would use speech and words to communicate with us? Furthermore, since we are the kind of being that exists in a sequential timeframe, and uses words to communicate between generations and pass knowledge down to those who come after, is it surprising that God would likewise reveal himself gradually over time? We may be slow learners when it comes to history, but we do eventually learn. These opening verses in Hebrews assert that God does use speech to reveal himself to us, and does so down through history. In the past, he used those men and women called "prophets", but finally he has used this being called "the Son".