How do we know that the world will end in the year 6,000? There are many differing contexts of the coming of the Messiah has anyone sorted out the contexts so that we can gauge a clear view?
Here are some sources for the world lasting six thousand years:
Talmud, Tractate Sanhedrin [97a]: “Rav Kattina said: ‘The world will exist for 6000 years and one (thousand) of destruction’ ... We have a teaching which is in agreement with Rav Kattina, as the seventh is the sabbatical year – one in seven years. Likewise the world will rest 1,000 in 7000 … ‘a thousand years in Your eyes are like yesterday which has passed …’ [Psalms 90:4].”
Talmud, Tractate Avodah Zarah [9a]: “The world will exist for six thousand years: two millennia of void (‘tohu’); Two millennia of Torah; and then two millennia of the age of Moshiach.”
Nachmanides, Commentary on the Torah (Genesis 2:3), quoting Kabbalistic sources: “Each of the Six Days of Creation corresponds to a 1,000 year period of future history.”
As for a clearer view of what it will be like when Moshiach comes, the basic answer is: a lot better than this.
From the Prophets, Talmud and Midrashim, we know what universal changes Moshiach will bring about. Among them: the whole world will return to God and His teachings; the entire Jewish People will be gathered back to the Land of Israel; the royal dynasty of King David will be restored; Jerusalem and the Third Temple will be rebuilt; the Temple service will be restored, along with the keeping of the Sabbatical Year (Shmitah) and the Jubilee Year (Yovel); the Sanhedrin, the religious supreme court of the Jewish people, will be reestablished.
No big deal, right?
In what order and in exactly what manner these events will occur, we do not know. G-d did not reveal the future that clearly to us. We do know certain ways that it cannot happen in any way which contradicts the Torah.
There are sources that do an excellent job of laying out clearly everything we do know about Moshiach. In English, the two best sources are The Handbook of Jewish Thought, Volume II, by Rabbi Aryeh Kaplan and The Way of G-d (Derech Hashem) by Rabbi Moshe Chaim Luzatto.