Assuming usual notation, I derive the floating rate and fixed rate payoffs and set them equal. The par swap rate I get thus is:
$$S_{mn}\mid_{t=0} = {\sum_{i=m}^{N-1} \tau_i L(0, T_{i-1}, T_i)Z_{0i} \over \sum_{n=m}^{N-1} \tau \cdot Z_{0i}}$$
When the $L$ and $Z$ curves are identical, the following can result as per the textbook - but I am not sure how?
$$S_{mn} = {Z_{0m} - Z_{0n} \over \sum_{i=0}^{N-1} \tau \cdot Z_{0i}}$$
If $L$ and $Z$ curves are identical, you are in a single curve frame work.
A swap can be seen as a long position in a fixed rate bond and a short position in a floating rate bond. (I'll use yearly payments 30/360 in order to be able to ignore the $\tau$ =1 and simplify the notation)
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n - (DF_1 \times C_1^{float} + ... + DF_n \times C_n^{float} + DF_n ) = 0$$
But because in a single curve framework (forward curve = discount curve) a floating rate bond with zero spread will always be at par, you have:
$$DF_1 \times C_1^{float} + ... + DF_n \times C_n^{float} + DF_n = 1$$
and so you only have to worry about the fixed leg:
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n - 1 = 0 $$
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n = 1 $$
If you rearrange this in terms of the fixed rate ($C^{fixed}$), you get:
$$C^{fixed} = \frac{1 - DF_n}{\sum^n_{i=1}DF_i}$$
Which is the formula you showed, just with a different notation.
The intuition of this formula is that you are determining the fixed rate on a bond where the sum of the present value of the interest payments plus the price of the zero coupon bond will be one.
You can also easily adjust this for a forward starting swap.
