75F - 201.5Fy = -201.5y + 111.5

"have we simply added 75F and subtracted 201.5Fy to/from both sides of the equation here?"
Yes, but not for the reason you stated. The entire purpose of this exercise is to solve for F, so we isolate it by moving both terms with F on the left side of our equation.

F(75 - 201.5y) = -201.5y + 111.5

"no idea what's happened at this point."
The reason we did the last step was so that we can pull out the common term F in order to solve for it. we are just dividing 75F and 201.5Fy by F, and then multiplying them by it. You can see that these expressions are equal by multiplying F times both terms.

F = (-201.5y + 111.5)/(75 - 201.5y)
Since our goal is to solve for F, after we isolate it we divide both sides of the equation by (75-201.5y) because this gives us an expression with F on one side and everything else on the other. This is called a function of F in terms of y.

so like to finish up the math, if our villain was 4betting QQ+, AK, and 78s, then folding just the 78s his fold F would be (4/38) (78s combos/all combos) and our y would be the pokerstove result of the equity of the hand we were considering against QQ+, AK.

One thing to note is that if villain is going to call your 5bet off with a relatively wide range (one including AQ) small PPs become the nuts due to their large equity against these ranges.