The Multiple Attack Penalty (MAP) rules on p446 state:
The more attacks you make beyond your first in a single turn, the less accurate you become, represented by the multiple attack penalty. The second time you use an attack action during your turn, you take a –5 penalty to your attack roll. The third time you attack, and on any subsequent attacks, you take a –10 penalty to your attack roll. Every check that has the attack trait counts toward your multiple attack penalty, including Strikes, spell attack rolls, certain skill actions like Shove, and many others.
The Basic Actions section on p469 states:
Many feats and other actions call upon you to use
one of these basic actions or modify them to produce
From this, we can determine that whenever a Feat or Action says "perform a Strike," we are actually performing the "Strike" basic action. The Strike basic action is what carries the Attack trait (note that Flurry of Blows does not).
Flurry of Blows p156
Make two unarmed Strikes. If both hit the same creature,
combine their damage for the purpose of resistances and
weaknesses. Apply your multiple attack penalty to the Strikes
normally. As it has the flourish trait, you can use Flurry of
Blows only once per turn.
From this, we can determine that spending the single action to perform Flurry of Blows results in the character performing two distinct Strike basic actions (with the MAP applied normally). That wording regarding MAP would not be meaningful unless that meant that it increased across the two Strikes.
After opening your turn with Flurry of Blows, you have performed two Strikes, so your next Strike has a penalty of -10. Note that unarmed attacks count as having the Agile trait, so the multi-attack penalty should be lower (a -8 for the third Strike in a turn).
The benefit of Flurry of Blows is that it's two Strikes in one action point, and that it sums the damage and applies resistance once across the two Strikes.