How to understand why the two period Differences-in-Differences estimator is the ATT estmator?

I read in a paper here that in a two time period differences-in-differences scenario where it claims the DiD estimator is the ATT (Average Treatment on Treated). I am trying to understand why that is. Denote the two time periods by $$t^*$$ and $$t^* - 1$$ and define a treatment indicator $$D_i$$, so that $$D_i = 1$$ for units that participate in the treatment and $$D_i = 0$$ for units that do not participate in the treatment.

Next, for $$t \in \{t^* - 1, t^*\}$$, define $$Y_{it}(1)$$ to be unit $$i$$'s treated potential outcome in time period $$t$$ (this is the outcome that it would experience if it were in the treated group), and define $$Y_{it}(0)$$ to be unit $$i$$'s untreated potential outcome in time period $$t$$ (this is the outcome that it would experience if it were in the untreated group).

The paper states that

$$\text{ATT} = E[Y_{t^*}(1) - Y_{t^*}(0) \mid D = 1]$$

There is an existing post here which discusses it, but I feel it fails to explain it intuitively.

It seems that in the DiD scenario, there is a treated and a control group at time $$t^*$$, since one got the treatment, and the other didn't. It therefore seems strange to talk about them being both treated. How can I think about this?