Give me a synopsis of the story as you understand it, then.
Big Boss, after a bitter spat with Zero, left The Patriots to found his own nomadic mercenary group, called MSF. Prior to the game, he met with Kaz who becomes his lieutenant and business manager of MSF. The KGB intends to hire him to drive CIA out of Costa Rica, and took the job because of a tape that potentially incriminates that The Boss is still alive that was taped by the friend of a girl who the KGB brought along, Paz.
Throughout his mission, he discovers that the CIA are bringing nukes into Costa Rica. He allies himself with the local rebels and expands the scope of his operations. He eventually discovers that everything was planned by a rogue CIA agent who was developing unmanned AI weapons to be the defacto nuclear deterrent force, using the engineering know-how of Hal Emmerich and AI expertise of Strangelove. Big Boss recruits Hal in the process.
Big Boss finds Strangelove, and discovers that The Boss's involvement in all this is because Strangelove, a precious person of The Boss's and also distraught of her death, took advantage of Hot Coldman's AI development offer to "recreate" The Boss in AI form, hoping to find out what her final will and to clear her name as a traitor. He eventually gets captured by Coldman/Strangelove, and Strangelove tortured him to find out the missing piece that was remaining to complete The Boss AI, which was her motivations for defection. Big Boss refused to tell the truth, but his almost irrational will to not change his story even in the face of torture only cemented to Strangelove that it was true all along that The Boss wasn't a traitor. (she already suspected it to begin with)
As Peace Walker was finished and moves towards the location where it would launch a nuke, BB chases it there but was caught once again. Coldman reveals that because of how PW works ( only retaliation ), he needs to upload fake data of an enemy strike before Peace Walker could launch its own nuke. He plans for PW's nuke to destroy MB, and that he was allied with the KGB all along in this whole charade. Unfortunately, the KGB backstabbed him and instead plans for the nuke to strike Cuba. Thankfully, MSF and the Sandisnistas manage to stop the KGB's plan, and the KGB agent was detained. Coldman suffered serious injuries in the skirmish though.
When everything seems like it was going to go over well, Coldman, in his last breath, uploaded false data of an incoming nuclear strike data to the Americans. Big Boss does everything he could, contacting the US to stand down, significantly damaging PW, removing the AI modules, but none of it seem to work at all. It appears like nuclear war was inevitable until at the very end, The Boss AI makes the conscious decision to drown itself to stop the transmission, thus preventing nuclear war.
At this point, Big Boss, who had long suspected it, but had been living in a state of emotional denial because he couldn't come to terms with the Boss's choice/way of life, decides that he cannot be the successor to the Boss's legacy, rejecting it and that he will be his own man, he will be above all those things, and that he will finally be "Big Boss."
Time passes as MSF continues to expand day-after-day, with the KGB agent detained and Paz under temporary protection of MSF. Throughout that time, Huey & Strangelove have slowly been building a Metal Gear for MSF itself, and Kaz recovered the nuclear bomb on Peace Walker and attached it to the Metal Gear. The KGB agent manages to escape a few times, but was captured at all times.
Eventually, it was discovered that the girl that the KGB agent brought along in the first place, Paz was in the fact a secret agent of Cipher/Zero, and that she had been observing behind-the-scenes the occurrences of everything since the very beginning. She reveals that Zero is offering Big Boss an opportunity to rejoin them once more, but with Big Boss's refusal, they have a fight atop Mother Base, of which Big Boss came out victorious and she fell into the sea.
Kaz later reveals that he knew all along that the Paz was actually an agent, and that his intention was to use her and the KGB agent's support to drastically expand MB, evident throughout the course of the PW mission. Of course, what he didn't tell Big Boss, was that he actually struck a secret business deal with Zero all along. The content and purpose of the deal is not made apparent to us, but he presumably did not know then what kind of a rabbit hole he was getting himself into.