What is the meaning of Abu Bakr's saying: "I am the successor (wali) of the Messenger of Allah, may Allah's peace and blessings be upon him," and Umar's saying: "And I am the successor (wali) of the Messenger of Allah, may Allah's peace and blessings be upon him, and the successor (wali) of Abu Bakr"?

The summary of the answer is:

The meaning of the Hadith is that Abu Bakr was the ruler (Wali al-Amr) after the Messenger of Allah, peace and blessings be upon him, so he dealt with the wealth left by the Prophet, peace and blessings be upon him, as the Prophet used to deal with it. Then Umar assumed authority after Abu Bakr, and he spent the wealth in accordance with what the Prophet, peace and blessings be upon him, and Abu Bakr used to do. Umar continued in this manner for a period of his caliphate. This is indicated by Umar's statement: "So Abu Bakr took possession of it and acted therein as the Messenger of Allah, peace and blessings be upon him, had acted... Then Allah caused Abu Bakr to die, so I became the guardian of Abu Bakr's affairs. I held it for two years of my leadership, acting therein as the Messenger of Allah, peace and blessings be upon him, had acted, and as Abu Bakr had acted therein."