You don't need to actually know the source material. In fact, pretty much everything F/SN and other series related is flavour.
There are going to be nine teams of two, and in order to win, you have to be the only remaining team.
When the game starts off, there usually is only a "Master" (Support person) and a "Servant" (Combat person). Sometimes the team can summon familiars, which usually are very weak helpers.
Servants go from Master to Master, but each Servant can only team up with one Master at a time. Each Master can have more than one Servant at a time.
Recently, there has been a surge of "alliances". Several teams group up in hopes of taking down individual teams, but be careful about who you trust as you might just get betrayed by the people who have your back.
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Feelin' the Super Saiyan Swag.