Throughout history, the survival of the Israelites in the face of their own severe failures stems from a profound divine restraint. God chooses to hold back His anger and avoid total destruction. This patience is not earned by the people's good deeds. Instead, it is driven by God's desire to protect His own honor, name, and glory in the world. Because God's name is permanently attached to the nation, wiping them out would lead to the disgrace of His reputation among the surrounding nations [רד״ק, אבן עזרא, מצודת דוד]. This divine motivation to act for His own sake extends to His glory as well [אבן עזרא, מצודת דוד]. While God's name represents His universal fame as the all-powerful Creator, His glory is revealed through His deep mercy. This glory is shown to the world through the very survival of a unique nation that remains attached to Him and eventually returns to Him even after committing serious wrongs [מלבי״ם, רד״ק].
According to strict justice, the people actually deserved immediate and total destruction in every single generation. This is especially true given that they sinned from their very beginning and remained unchanged despite witnessing great miracles [מלבי״ם, רד״ק]. Yet, God delays His punishment. This patience is compared to a person who calms their anger by taking long, slow breaths, thereby putting off the urge to strike [ביאור שטיינזלץ].
To illustrate this restraint, vivid imagery is used. The primary approach among commentators connects this self-control to the physical idea of a nose. In ancient poetic terms, fierce anger is pictured as hot smoke pouring out of a person's nostrils. God essentially declares that He will plug His nose to trap the smoke of His fury, preventing it from escaping and harming the people [רש״י, מצודת דוד]. Others understand this concept as a muzzle or a barrier, meaning God actively restrains and blocks Himself from delivering a harsh penalty [שד״ל, ביאור שטיינזלץ]. Even when God is forced to bring about consequences, He holds back a portion of His anger so that the resulting damage is only partial [מלבי״ם].
The ultimate goal of all this intense self-restraint is simply to keep the nation from being entirely wiped out, which is the fate they technically deserved. Historically, this dynamic played out during the era of the First Temple. Rather than destroying the Israelites completely for their actions, God chose the lesser consequence of exiling them to Babylon, ensuring their continued existence [שד״ל, מצודת דוד].