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Inspired by this question.
The age when one is obligated to keep Mitzvos is 13 (for a male and 12 for a female). (See Avot 5:21 and Bartenura)
After that age, Beit Din may punish them if they transgress the law.
Punishment from heaven starts at 20 years old (Shabbat 89B).
Why is the age when one may be punished by Beit Din 13 years old, and the age of punishment from heaven 20 years old?
The Chacham Tzvi (Paragraph 49) gives 4 answers about whether punishment starts at 13 or 20 years old (brought in the Pardes Yosef here):
It was only before the giving of the Torah that punishment started at 20, after the giving of the Torah it starts at 13. (I believe this only answers why we say Sarah was free from sin at 100 as when she was 20. It does not address the Gemara or any others who say that Heavenly punishment starts at 20)
The Heavenly Court doesn't always punish for crimes committed under 20, and that's part of Yitzchok's "deal" with G-d, to ignore any sin committed before 20.
The Heavenly Court doesn't punish people in this world for doing something wrong when they are under 20. Once they go up to heaven, however, they are judged for everything done after 13 years old.
The Heavenly Court waits until one is 20 to see if they do Teshuva. Once the person reaches 20 without doing Teshuva, the person is punished for the sin he committed when younger.
The Pardes Yosef on Parshat Chayai Sarah (Chapter 23) gathers many different opinions about what it means "The Heavenly Court does not punish someone under 20 years old". It appears he includes most (if not all) of the opinions brought here (and more), as well as giving one of his own.
The Pardes Yosef himself (I think) points out that heavenly punishment is stricter than earthly punishment, as is brought in Baba Kamma (55B-56A). The Gemara there brings cases where the earthly Beit Din would not punish for actions (since they are too inconsequential), but the Heavenly Court will still extract punishment. When it comes to time, however, the Heavenly Court is more lenient, waiting until 20 to punish.
This is because due to the public nature of the punishment. When the earthly Beit Din punishes someone, it is for their own benefit, since they will learn not to do it again. Even when he is killed by Beit Din, other people learn from this, and refrain from sinning themselves. This is good for others, and good for the punished as well, since this merit helps his atonement process.
On the other hand, when a person is punished by the hands of G-d, it is too late for him to learn from his punishment, and therefore G-d waits till the person is 20 before holding him liable for his actions.
Furthermore, the Pardes Yosef continues, it says in Sefarim that when a person sins he causes a blemish on his soul, and Heavenly punishment wipes away these blemishes. A person under 20 has not yet achieved full intellectual maturity, and therefore the blemishes are not so great and do not yet require cleansing. (He brings a proof from our Sages that a person does not reached full intellectual maturity until 20, but I am unfamiliar with the statement and he does not bring a source, so I can't look it up.)
Here Rabbi Zweig brings a different explanation in a similar vein.
I heard a shiur by Rabbi Uziel Milevsky who explains as follows.
the earth beis din punishes the ACT only. when a person reaches 13 he is a man and therefore his act is a full act.
the heavenly beis din looks also at the motives, maturity level, etc. and since a person does not fully know what he is doing until 20 the he is exempt until then
tl;dr: Either a) this applies only to negative mitzvos, which someone under twenty is more susceptible to break, so Hashem gives people under twenty a free pass, or b) this applies only to mitzvos sichlios, those that an intelligent and mature mind can determine are appropriate, so those under twenty are less likely to come up with them on their own and are not held accountable.
To further @Menachem and @ray's answers:
The Chacham Zvi (Simman 49) asks two questions on the gemarra in Shabbos 89b that implies the Heavenly Court doesn't punish for sins comitted under twenty:
Milah, the Rambam (Hilchos Milah 1:2) and Raavad (ad. loc). sound like once a person turns thirteen and doesn't perform milah, they get kares.
Pesach, the Torah says (Numbers 9:13) that an ish that doesn't perform the Pesach get's kares. Ish throughout the Torah means someone who is thirteen and above.
The Magen Giborim (Orach Chaim 219, Shiltei Giborim 1) wants to answer these two questions (although he agrees with the Chacham Tzvi's conclusions) as follows: Those are both positive mitzvos. We can differentiate between positive and negative mitzvos.
Positive mitzvos even if they are broken above the age of thirteen have no earthly punishment. As such, they shouldn't get off scot-free, and get heavenly punishment.
Negative mitzvos involve succumbing to one's baser desires, and we can say someone under the age of twenty is more susceptible to giving in to his inclinations. Therefore, in Heaven they let earthly punishment suffice.
However, Rav Dovid Kronglass zt"l, Mashgiach of Ner Yisroel (Sichos Chochmah UMusar #50) brings more questions on this gemarra: Why is someone between 13-20 whose ox kills another obligated to pay kofer, which is to prevent death by Heaven1. He also asks from the children from Sedom and the dor haMabul why they were punished. The explanation of the Magen Giborim isn't sufficient.
Based on these questions, Rav Kronglass concludes the approach of the Chavos Yair (#166) and Chasam Sofer (Yoreh Deah #155) is correct, that the less than twenty idea applies only to mitzvos sichlios, meaning only to mitzvos that an intelligent and mature mind can determine are appropriate2. This explains Sedom and the dor HaMabul, who broke explicit prohibitions (gezel, arayos).
I don't remember if he says this explicitly, but it would then follow that the age of twenty is determined because until that age a person can't be expected to be mature enough to realize what should and shouldn't be prohibited on their own.
1 He proves they're obligated from Bava Kama 40a which states a child is exempt and the Rambam (Hilchos Nizkei Mammon 10:6) who says children, deafmutes and deranged people are exempt due to not being obligated in mitzvos (לאו בני חיוב). Therefore, a child who is obligated in mitzvos (13-20) is obligated in kofer.
2 Cf. the Chidah (Nachal Kadmonim parshas Chayei Sarah #2) who says that the Chacham Zvi's approach isn't the simple reading of the gemarra, and I would venture to say he would say the same about this approach.
First, note that this notion of not being punished from heaven until 20 is a midrash here, not part of the Halacha.
Rashi on that passage in Shabbat 89B linked in the OP's question reminds us of the source for this notion: in the dessert, the people that were punished for the חטא המרגלים were only those 20 and up (see Numbers 14:29 בַּמִּדְבָּר הַזֶּה יִפְּלוּ פִגְרֵיכֶם וְכָל-פְּקֻדֵיכֶם, לְכָל-מִסְפַּרְכֶם, מִבֶּן עֶשְׂרִים שָׁנָה, וָמָעְלָה: אֲשֶׁר הֲלִינֹתֶם, עָלָי. )
It appears that this source is used here (= in Shabbat 89B) as part of the common Midrash Pasuk method to support the beautiful prosaic "negotiation" between Yitzhak and Hashem about the magnitude of Israel's sins (70 years of life, of which only 50 are punishable, of which 25 are nights, of which half are prayers, food, etc. - leaving only 12.5 which "we can split half ways") - and not meant as a source for e.g., Halachic discussion.
