By the above authentic tradition...
By the above authentic tradition, the Prophet (PBUH&HF) meant that as Moses had left behind Aaron to look after his people when he went to Miqaat (meeting Allah), in the same way he was leaving Ali behind to look after the affairs of Islam after he met Allah (i.e., his death).
Confirming what the above tradition implies, we find in the many reports that Imam Ali (AS) received the title of the "brother" of Prophet when Prophet established the "brothering" among his followers (see Sahih al- Tirmidhi, v5, p363; Sirah Ibn Hisham, p504; Tahdhib al-Tahdhib, v4, p251). Interestingly enough, the Prophet in that occasion made Abu Bakr and Umar brother of each other (al-Tabaqat, by Ibn Sa'd, v3, part 1, p123).
If Abu Bakr was really the closest to the Prophet, he would have chose him for himself instead of Imam Ali. In fact, if we look deeper to the situation of after death of Prophet Mohammad (PBUH), and the leaving of Moses to MIQAAT (appointment with Allah), we will see more analogy to what Prophet (PBUH) said to Ali (AS).
Quran states that: Moses (AS) with the order of Allah, assigned Haroon (AS) as his successor (Caliph) and left his people to him, and left for MIQAAT (appointment with Allah) for a total of forty days. After leaving of Moses, most of his companions turned against Haroon, and were deceived by Sameri, and became worshipers of a golden calf. (See Quran 7:142, 20:90-97, 20:83- 88). The analogy that Prophet (PBUH) mentioned in the above tradition, seems to be a reality after his demise.
Most of companions (except Abu Dhar, Miqdad, Salman al-Farsi, Ammar, and ...) became disloyal to Ali (AS) after the death of Prophet (PBUH), turned against him, and preferred some other people to him. The majority of people disobeyed Ali (AS), as their forefathers disobeyed Haroon (AS). They did not take lessons from Quran and the history, and thus the history repeated over and over again.
The repetition of the history of the Children of Israel for Muslims is confirmed by Prophet (PBUH&HF): Sahih al-Bukhari Hadith: 9.422 Narrated Abu Sa'id al-Khudri: The Prophet said, "You will follow the ways of those nations who were before you, span by span and cubit by cubit (i.e., inch by inch) so much so that even if they entered a hole of a mastigure (lizard), you would follow them." We said, "O Allah's Apostle!
(Do you mean) the Jews and the Christians?" He said, "Whom else?" This tradition is also narrated by Muslim in his Sahih , v8, p57.