There's not a super-big history to this, beyond it being a Persian-derived suffix in Persian that "land" or "place". It's roughly analogous to the "-ia" ending from Latin that similarly gets attached to place names.

Other languages actually call loads of countries names ending in -stan: in Turkish you have Macaristan, Yunanistan, Bulgaristan and Sirbistan for Hungary, Greece, Bulgaria and Serbia, respectively. Persian similarly has such uses, as well as names like Hindustan for India, which actually gets used a fair amount in Indian contexts. Armenian uses the suffix for country names as well, referring to itself as Hayastan (in Turkish it's Ermenistan, and Persian has a similar name).

As for specifically country names in English, it's actually a relatively new phenomenon. For quite a while the only country with such an ending was Afghanistan. While formed from areas such as Waziristan and Balochistan, Pakistan of course only dates to 1947, and the name itself was coined in 1933.

The five former Soviet "stans" are even more recent. In Soviet times such names were unofficial, and their official designations were "Kazakh/Uzbek/Kyrgyz/Turkmen/Tajik Soviet Socialist Republic". Even in unofficial uses occasionally "ia" endings from Russian were more commonly used, especially "Kirgizia" and "Turkmenia". These five republics only officially switched to the "stan" designation in 1991, and even in the case of Kyrgyzstan it's still an unofficial name, as officially the country is the Kyrgyz Republic.