The Jews believe in the Written Torah as it was given word for word from God to Moses. But additionally, we believe that Moses was taught by God an Oral Torah, that is, a tradition of how to explain the text of the Written Torah, how certain laws are applied, how we practice certain mitzvos, and other additional concepts pertaining to Jewish law.
Moses taught this oral tradition to the Jews and the leaders of the nation, and it was passed down generation to generation until it finally came time to write it down lest it be eventually forgotten or distorted, which resulted in the Mishna, and then later expounded upon in the Talmud. The Talmud as we know it is a collection of rulings and legal discussions with regard to not only biblical laws, but rabbinic laws as well. But when the Talmud discusses biblical law, it assumes the existence of the oral tradition. It tells of the laws as they are traditionally understood through the lens of the Oral Torah.
There certainly is no lack of history when it comes to many Jews denying the existence of the Oral Torah. The Sadducee people come to mind, as well as Spinoza. But in general, the argument usually given for the authority of the Oral Torah is the obvious ambiguity of the laws of the Written Torah when taken at face value. Here are a couple classic examples:
- One of the laws of the Torah is the daily donning of tefillin. But the Torah makes no mention to how this is done, to what tefillin are, to what goes inside them, nor to whom this applies (men, women, children?) Only the oral Torah addresses any of this.
- The Torah writes that one may not do work on Shabbos. But it never mentions what constitutes work. Only with the Oral Torah is it clearly defined how one transgresses Shabbos.
It seems pretty clear that some form of tradition must exist to observe these mitzvos that we would otherwise not have any idea what they are talking about.
Another argument is simply that it was given to us by the previous generations along with the Written Torah. In other words, just as we believe the previous generation that this Written Torah they are showing us is divine, we believe their interpretation of it as well.
A story illustrating this argument is related in the Talmud (Shabbos 31a):
A non-jew who believed in the Written Torah wanted to convert. He went to the great rabbi Shamai and asked him, "How many Torahs do you have?". Shamai responded, "Two. One written and one oral." The gentile said, "I wish to convert to Judaism, but on the condition only to accept the written Torah." Shamai sent him away. Then the gentile went to Hillel and said again, "I wish to convert to Judaism, but on the condition only to accept the written Torah." Hillel accepted him as a student for conversion. The first day Hillel taught him the Hebrew alphabet. When he came the next day, he reviewed the alphabet, but this time, Hillel switched the letters, that is, he referred to letter aleph as "beit", and beit as "aleph", etc. The non-jew protested, "But yesterday you taught me differently!" Hillel responded, "You see? You rely on me for your understanding of even the simplest thing like the names of the letters."
Hillel's point was that we have no idea what the Torah means without the way it was taught to us and explained to us by our Rabbis. This was his reasoning for the authenticity of the Oral Torah.