The fifth part of du'a starts with a verse from the Chapter of the Spoils of War, in the Quran. According to tradition, this verse was revealed after a man named Abu Lubaba betrayed Muhammad to a Jewish Tribe, Banu Qurayza. Apparently, the verse was so powerful that Abu Lubaba starved himself until the Prophet forgave him (after about a week). So, followers of the Prophet were specifically instructed not to break the trust of the Prophet, and by extension his progeny and the Imam.
At the pond in Khumm, the Prophet placed his trust in the unity of the "Book of God" and the "People of the House". As an extension of this trust, Ismailis believe that the Quran cannot be properly understood without its counterpart, the Imam. According to belief, only the Imam knows the true meaning of the Quran and it's his job to disseminate that information to his followers. This verse the the du'a reiterates the trust that Ismailis have in their Imam (and in God, through the Imam) for this interpretation.
Remembrance (ذِکْر "dhikr" in Arabic) is an important part of Islam, in general. It's even mentioned in the Quran that Muslims should remember God in the morning and evening. Coincidentally, these are the times that Ismailis are instructed to recite their Du'a. Tasbih (تسبيح in Arabic) is a form of remembrance where a name of God or short praise of God is recited repetitively. Ismailis also call the device used for counting these repetitions a "tasbih". You can read more about tasbihs here.
In some beliefs, Muhammad and 'Ali are considered "living" names of God. This is because of their exalted status as messengers of God and leaders of the community. According to Ismaili scholar Mu'ayyad filDin alShirazi, Muhammad declared: "I and 'Ali are the parents of believers." It's important to note that calling these names during Du'a is not considered worship of Muhammad and 'Ali, but worship of God through Muhammad and 'Ali.
After the quiet recitation, the du'a continues by emphasizing the support and reliance Ismailis have on the current Imam. This comes from the belief that the Imam (as an extension of Imam 'Ali, and alShirazi's above consideration) acts as a spiritual parent to Ismailis, and thus provides support to the community. This is also in addition to the reliance on God in the first part of du'a.