How to say the du'a Sayyidul Istighfar? What does it mean?

Prophet Muhammad (ﷺ) said that the most superior request for forgiveness is to say,

"Allahumma Anta Rabbi, Laa ilaha illa anta"

"O Allah, You are my Lord. There is no other God besides You." Here, we have started off by saying, Allahumma, O Allah, You are my Lord, we are accepting recognition of Allah's Dominion over all of our affairs. There is no God besides You. There is none to be worshipped in this world, inside or outside, besides You.

“Anta khalaqtani wa ana abduk wa ana ala ahdika wa wa’dika ma istata't,”

“You created me and I'm Your servant following my covenant and my promise to You as much as I can.”  Recognizing that Allah is the one who created us and He is the one we will return to. When He created us, a covenant was made. As He has explained in Surah al A'raf, "And when He created Adam, He took from Adam all of his descendants, and He made them bear witness saying,

‘A lastu bi rabbikum (Am I not your Lord)?’  Then we all bore witness that He is our Lord."  [1]

That is the covenant, the covenant to worship Him in this life, and our promise to do it  as much as we can.

“A'udhu bika min sharri maa sana't,”

“I seek refuge in you from the evil of what I have done.” This is a du'a, a supplication that we are asking for Allah's forgiveness. So, we have to start off by focusing on what it is that we are asking forgiveness for. It's for an evil which you have done, or all the evil which we did. Then, the Prophet (ﷺ) told us to say,

"abu'u laka bi ni'matik alayya, abu'u bi dhanbi"

“before You, I acknowledge Your blessings bestowed upon me and I confess my sins to You.”  I recognize in my life that You have blessed me, there is no excuse for my sins, and I confess my sin. I admit it before You.

“Fagfir li fa innahu laa yagfiru dhunuba illa anta,”

“so forgive me for indeed no one forgives sins except You.” We ask Allah ultimately to forgive us for our sins, because no one else can forgive us. If we have done something wrong to someone else, we could ask them to forgive us, but it does not take the place of seeking forgiveness of Allah. Though people in this life may forgive us, in doing whatever evil we had done to them; we have not only harmed them but we have harmed ourselves by disobeying Allah's command. We have to turn to Allah and seek Allah forgiveness. Ultimately, we recognize that it is only He who could forgive.

Prophet Muhammad (ﷺ) said in the end of that, "Anyone who says that, firmly believing in it and dies before the evening, or he/she says it in the evening, firmly believing in it and dies before the next morning, he/she will be from amongst the people of paradise." [2]

References:

[1] [Qur’an 7:172]

[2] [Sahih Al-Bukhari:  Book 20, Hadith 1875]