Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates
$T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the payments $L(T_{i-1}; T_{i-1}, T_i)\Delta T_i$ and $K\Delta T_i$, where $\Delta T_i = T_i -T_{i-1}$, and
\begin{align*}
L(t; T_{i-1}, T_i) = \frac{1}{\Delta T_i}\bigg(\frac{P(t, T_{i-1})}{P(t, T_i)}-1 \bigg),
\end{align*}
for $i=1, \ldots, n$, is a forward Libor rate. Here, $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face value.
The value of the swap at time $t$, where $0 \leq t \le T_0$, is given by
\begin{align*}
& \ \sum_{i=1}^n \frac{1}{\Delta T_i}\bigg(\frac{P(t, T_{i-1})}{P(t, T_i)}-1 \bigg) \times \Delta T_i \times P(t, T_i) - K \sum_{i=1}^n P(t, T_i) \times \Delta T_i \\
= & \ P(t, T_0)-P(t, T_n) - K \sum_{i=1}^n P(t, T_i) \Delta T_i\\
= & \ P(t, T_0)- \bigg(\sum_{i=1}^{n-1}K \Delta T_i P(t, T_i) + \big( 1+ K \Delta T_n\big) P(t, T_n) \bigg).
\end{align*}
The swaption payoff at maturity $T_0$ is given by
\begin{align*}
& \ \Bigg[\sum_{i=1}^n \frac{1}{\Delta T_i}\bigg(\frac{P(T, T_{i-1})}{P(T, T_i)}-1 \bigg) \times \Delta T_i \times P(T, T_i) - K \sum_{i=1}^n P(T, T_i) \times \Delta T_i\Bigg]^+ \\
= & \ \Bigg[1- \bigg(\sum_{i=1}^{n-1}K \Delta T_i P(T_0, T_i) + \big( 1+ K \Delta T_n\big) P(T_0, T_n) \bigg)\Bigg]^+.
\end{align*}
That is, the swaption payoff is the payoff of a bond option that has a coupon rate the same as the swap fixed rate, while the bond option strike is 1.