"It does not say that the purpose of creation of this world is as an arena for man's free will." That is correct. However, that is not what you paraphrased from the lecture, i.e., the purpose of nature.
Where does it say this? In the word itself! The word teva means coin, meaning the same thing in Hebrew as it does in English. A coin is something agreed upon and stable, both in monetary notes and metal, and in terms and natures. Hashem runs the world. When we see the same thing happening over and over again, however, it becomes plausible to ascribe happenings as having their own existence. This thought is only a few hundred years old, however. Before then, nature was attributed to some deity or force, either directly or indirectly. If a verse were required for the Jewish perspective, it would be Bereishis 1:1.
There is another implied question in your post, however. That is, why did Hashem create the world? That question is not answerable, and is, according to Rambam Avodah Zora 2:5-6 from Bamidbar 15:39, forbidden to spend time thinking about. Hence, we cannot say if it was for free will or not. Indeed, it is unlikely, as Avos 4:21 compares this world as being secondary to the next world. It is the next world that is more likely to embody or contain the purpose of creation.
Nonetheless, we can still infer certain intentions contained within the creation. 1 Shmuel 15:22 has Shmuel HaNavi lecturing to Shaul that listening, rather than action, is what Hashem wants. Avos 5:1 explains that Hashem created the world with 10 sayings rather than 1 in order to payback the rasha and to give good reward to the righteous. Midrash Rabba 1 mentions Gan Eiden and Gehinom many times as being part of the creation.
If this is what Hashem wants, affected how He created the world, and was part of the first days and declared good, we can infer that it is an important part of this world.
