How would the actions of the Great Purge increase Stalin's power?

The Great Terror atomized Soviet Society and made even the idea of any opposition to Stalin impossible.

It is difficult for those of us living in freer societies than that of the USSR in the 1930s to truly understand the effects of such ruthless government terror like that initiated by Stalin in the Purges. Writers like Arthur Koestler and George Orwell tried to explain, historians like Robert Conquest and Alexander Solzhenitsyn charted the effect. Even today, Russians really don't like to examine the subject much.

Essentially, Stalin concocted the idea that there was a 'plot' against the Soviet Union (the 1934 murder, probably at Stalin's behest of Sergey Kirov), investigated the 'plot' to determine that there were agents working for Trotsky and other external foes to weaken the USSR, and then started finding 'traitors' and punishing them brutally.

These 'traitors' were variously described as the "Trotskyite-Kamenevite-Zinovievite-Leftist-Counter-Revolutionary Bloc" and "anti-Soviet Trotskyite-centre", and then just "Trotskyite wreckers". With such indistinct crimes, anybody could be arrested for anything and tortured into confessing anything. Often, officials were expected to produce a quota of suspects and soon found themselves -- to save their own necks -- zealously exceeding their quota. For most, trial was unnecessary, and a gulag sentence or mass execution was their fate.

The search for these supposed traitors allowed Stalin to purge the Communist Party of its oldest members; eliminate all potential rivals; and intimidate both his military and the NKVD (twice purged themselves to get rid of anyone with an inconvenient memory).

Party members and many other Soviet citizens had no idea of what the 'rules' were for survival, except that a loudly expressed devotion to Stalin seemed a safe practice (and wasn't always). But you express that love often enough and it tends to become real to you. A lot of people also became complicit -- denouncing others before being denounced themselves. Guilt kept many Russians quiet for a long time.