Making sense of the delay formula between two sensors in an antenna array?
but I don't see how the quantity 'dsin(theta)' represents the extra distance


Let analyse just sensors 1 and 2 also put $\theta$ 90 and 0 degrees and see what happens.

Let $\theta$ = 0 degrees, which means that transmitter of the signal is situated in front and both sensors should receive the signal at the same time (isn't it?).

$$
\Delta_i = { {d \cdot \sin {\theta} } \over {c} } = { {d \cdot 0 } \over {c} } = 0
$$

Let $\theta$ = 90 degrees, which means that transmitter of the signal is situated at the right hand and first sensors should receive the signal later. And delay should be time which need for signal travel the distance d with velocity c.

$$
\Delta_i = { {d \cdot \sin {\theta} } \over {c} } = { {d \cdot 1 } \over {c} } = {d \over c}
$$

But anyway in my opinion the indexes of sensors should be in the opposite direction.

I hope it helps you.