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.