Forward rates are the implied, no-arbitrage floating rate in the 'future'. I use this term loosely here because nobody really pretends that these rates are actually realized in the future, but, mathematically, forward rates are implied fundamentally as:
The rate that would make you indifferent between locking in an interest rate from t0 to t1 and then rolling into another (pre-decided rate) between t1 and t2, or just one, "spot" rate from t0 to t2 outright.
Mathematically:
\begin{equation}
(1+f_{0,1})*(1+f_{1,2}) = (1+r_{0,2})
\end{equation}
(note that i ignored compounding and conventions to show the simple case, so this assumes each forward is only compounded for the single period and day-counts perfectly match)
where technically
\begin{equation}
f_{0,1}=FloatingRate_{0}
\end{equation}that is, the first forward rate is the first/current known floating rate.
The implied forwards of the current floating rate are used for pricing the legs of the swap to get 0 NPV (assuming vanilla swap). If you have a full forwards curve, or a full zero curve, then you can simply solve for the other (assuming the times align, if not you have to use what is called 'stub rates', but won't get into that).
When it comes to swap value, qualitatively the link between floating and forward rates are simply: if the floating rate does not follow the path of the initial implied forwards (at entry of the swap contract), then the value of the swap will change. At first this statement may seem strange, because often swaps are viewed as a bet on the fixed rate leg and that rate moving up and down. But think about how the swap is valued. In theoretical pricing we solve for the fixed leg (in practice you may find yourself solving for a spread to the float leg, but won't get into that here). So what actually causes the fixed leg to change? Well, if the zero curve moves (or par curve on which zero rates are implied) then now your implied forwards have changed! Obviously now your swap has changed in value from the 0NPV entry point (ok... it's possible things shift in ways that it stays near 0... but you get the point). So in reality it makes more sense to think about a swap as a bet on implied forward rates, but nobody really does because it is easier to measure in dollar value of basis point terms on the fixed leg.
If I understood your question this should help establish the relationship between floating and forward rates. I would point out that when I say "0NPV" it is not because the PV of each legs is zero, but rather the difference between the two is zero, so that in aggregate the two legs cancel out in PV upon entry.
Forwards and how they relate to discount factors (which are present values of zero coupon bonds. I admit some of your subscripts confuse me a bit just because I'm not used to seeing it that way, but it looks correct to me so hopefully you can relate the following back to your notation):
\begin{equation}
(1+f_{0,1})(1+f_{1,2})=(1+r_{0,2})
\end{equation}
\begin{equation}
1/(1+f_{0,1})*1/(1+f_{1,2})=1/(1+r_{0,2})
\end{equation}
\begin{equation}
where: 1/(1+r)=DF
\end{equation}
\begin{equation}
DF_{0,1}*1/(1+f_{1,2})=DF_{0,2}
\end{equation}
\begin{equation}
DF_{0,1}/DF_{0,2}=(1+f_{1,2})
\end{equation}
I purposely left the forward rate because this is often how I've seen the relationship stated, however if we take 1/ both sides, it is equivalent to:
\begin{equation}
DF_{0,2}/DF_{0,1}=DF_{1,2}
\end{equation}
but \begin{equation} DF_{1,2}\end{equation} doesnt really have much economic intuition (it's a 'future present value' so to speak) in practice, so it is avoided, but we can still reconcile to your formula in the comments:
Firstly, lets get this out of the way:
\begin{equation}
DF_{0,t}=P_0(t)=1/(1+r_t)
\end{equation}
In the general case where we allow for different time periods we then have:
\begin{equation}
P_0(t)/P_0(t-1)=P_{t-1}(t)
\end{equation}
simple rearrange:
\begin{equation}
P_0(t)/P_{t-1}(t)=P_0(t-1)
\end{equation}
where i've used t instead of pi, which IMO is less confusing, and I think the formula you have in the comments erroneously uses both.
