Your answer requires more space (and more knowledge) than I have here, but I'll give it a shot, using Rambam's introduction to the Mishnah. (But please read the whole Introduction if you can.)
Some of the statements of Rabbis are explaining laws that were revealed to Moses as part of the Oral Law. Some of them also have sources in verses that could be used to show that the law is true, and some were just accepted as tradition.
As the Rambam writes
דע כי כל מצוה שנתן הקב"ה למשה רבינו ע"ה נתנה לו בפירושה. היה אומר לו המצוה ואחר כך אומר לו פירושה וענינה.וכל מה שהוא כולל ספר התורה.
Know that each commandment that the Holy One, blessed be He, gave to Moshe, our teacher – peace be upon him – was given to him with its explanation. He would say to him the commandment and afterward tell him its explanation and content; and [so too with] everything that is included in the Book of the Torah.
Fortunately for us, the Rambam's case used for illustration is Sukkah.
והנה לך משל שהקב"ה אמר למשה בסכות תשבו שבעת ימים (ויקרא כג):
And behold for you an example: When the Holy One, blessed be He, said to Moshe, "You shall live in booths seven days" (Leviticus 23:42);
The Rambam then goes on to give the specific example you asked for:
והודיע שהאכילה והשתיה והשינה בה כלו חובה. ושלא יהיה בחללה פחות משבעה טפחים אורך על שבעה טפחים רוחב. ושלא יהיה גובה הסוכה פחות מעשרה טפחים.
And He made known that eating and drinking and sleeping in it is all an obligation; and that its space should not be less than seven finger-breadths long on seven finger-breadths wide, and that the height of the booth not be less than ten finger-breadths [sic, should be handsbreadths -N.T.].
The Rabbis also had the ability to add decrees to protect the original biblical law. The Rabbis have the authority to do this from the verses in the Torah telling the Rabbis to guard the Torah. As the Rambam also explains there
והחלק הרביעי הם הגזרות שתקנו הנביאים והחכמים בכל דור ודור כדי לעשותם סייג לתורה. ועליהם צוה הקדוש ברוך הוא לעשותם והוא מה שאמר במאמר כללי (ויקרא יח) ושמרתם את משמרתי ובאה בו הקבלה (יבמות דף כא.) עשו משמרת למשמרתי. והחכמים יקראו אותם גזרות.
And the fourth division are the ordinances that the prophets and the sages ordained in each and every generation, in order to make a fence around the Torah. And about them did the Holy One, blessed be He, command to do them, and it is what He said in the general statement (Leviticus 18:30), "And you will guard My guarding" – and the received tradition came [to explain] (Yevamot 21a), "Make a guarding around my guarding." And the sages called them ordinances.
The Sages were also given authority over financial laws, including to remove money from one person and give it to another, which is called Hefker beis din hefker. As the Rambam explains:
וְכֵן יֵשׁ לַדַּיָּן תָּמִיד לְהַפְקִיר מָמוֹן שֶׁיֵּשׁ לוֹ בְּעָלִים וּמְאַבֵּד וְנוֹתֵן כְּפִי מַה שֶּׁיִּרְאֶה לִגְדֹּר פְּרָצוֹת הַדָּת וּלְחַזֵּק הַבֶּדֶק אוֹ לִקְנֹס אַלָּם זֶה וַהֲרֵי הוּא אוֹמֵר בְּעֶזְרָא (עזרא י ח) "וְכל אֲשֶׁר לֹא יָבוֹא לִשְׁלֹשֶׁת הַיָּמִים כַּעֲצַת הַשָּׂרִים וְהַזְּקֵנִים יָחֳרַם כָּל רְכוּשׁוֹ" מִכָּאן שֶׁהֶפְקֵר בֵּית דִּין הֶפְקֵר:
Similarly, at all times, a court has the prerogative to declare money belonging to others as ownerless. It may destroy those funds or give them to whomever they see fit to close any breaches in the faith and to strengthen its observance or to penalize a stubborn and difficult person. The Book of Ezra 10:8 states: "Whoever fails to come in three days according to the advice of the officers and the elders will have all of his property confiscated." From this we learn that when a court declares property ownerless, their declaration is effective.
Even if [its value] appreciated in the hand of the robber, the appreciation is to the one who was robbed. And this is [the understanding] of that which is written in the Torah, "and return the theft that he robbed." And the explanation comes - if it is the same as what he robbed, meaning that it has not changed, he must return it as it is, and even if it appreciated much.
But the Sages ordained (Bava Kamma 94b), as a result of the Ordinance of the Penitents, that anything that appreciates in the hand of the robber after the forsaking be his. And [so] when he comes to return it, he calculates with the robbed one how much it was worth at the time of the robbery; and the robbed one pays him the money of that which it appreciated, and takes it. And the Sages have the power to do this thing, since they may do according to their will in a monetary matter - and even against the commands of the Torah. As it is well-known that what the court makes ownerless is ownerless (Yevamot 89b, Gittin 36b).