What is the explanation of the Hadith: "The Prophet (peace and blessings of Allah be upon him) used to breathe into the vessel three times"?

The tradition mentioned in the question has an interrupted chain of narration (غير متصل الإسناد). It suggests that the Prophet (peace be upon him) paused three times during the recitation of Al-Fatiha, which contradicts the more authentic accounts stating that he (peace be upon him) would pause at the end of each verse. This is what scholars have affirmed to be the Sunnah for the reciter. The proof for this is that Umm Salamah (may Allah be pleased with her) was asked about the Prophet's (peace be upon him) recitation, and she said: "He would cut his recitation, verse by verse: 'In the name of Allah, the Most Gracious, the Most Merciful. All praise is due to Allah, the Lord of the worlds. The Most Gracious, the Most Merciful. Master of the Day of Judgment.'" This was narrated by Imam Ahmad. Ibn Al-Qayyim mentioned that the Prophet (peace be upon him) would pause after "{Lord of the worlds}", then begin "{The Most Gracious, the Most Merciful}" and pause, then begin "{Master of the Day of Judgment}" with deliberation, slowness, and clear articulation (tartil).