The simple reason is that Chechnya was never a "Soviet Socialist Republic" of the Union of Soviet Socialist Republics. Instead, Chechnya was an "Autonomous Soviet Socialist Republic" of the Russian Soviet Federative Socialist Republic, which was the Republic that represented Russia in the USSR.

When the USSR fell apart, only it's constituent SSR's seceded from the USSR. In order for Chechnya to become an independent republic, it needs to secede from the RSFSR, and the RSFSR did not allow it's ASSR's to secede.

Edit : To clarify the jumble of Acronyms

USSR: Union of Soviet Socialist Republics, the collective of 15 Soviet Socialist Republics (SSR). AKA the Soviet Union. Cold war references to "Russia" are usually referencing the USSR.

RFSFR: Russian Soviet Federative Socialist Republic, the largest and most powerful of the SSRs in the USSR, and the one that would eventually become the modern Russian Federation. When the Soviet Union broke up, the RSFSR seceded from the Soviet Union.

Chechnya: The geographical region of Chechnya. It was part of the Chechen-Ingush Autonomous Soviet Socialist Republic (ASSR). The Chechen-Ingush ASSR was a part of the RFSFR, and after the fall of the Soviet Union, re-organized as a Republic of the Russian Federation (but not before much bloodshed).