Suppose the original TWFE equation was something like the following: $$ y_{nt} = \alpha_n + \mu_t + d_{nt}\beta + u_{nt} $$ where $n$ is the index for units (e.g., individuals, firms) and $t$ indexes time. $\alpha_n$ is the usual unit fixed effect, while $\mu_t$ is an intercept for the time period (often called a "time fixed effect," but some people don't like that term). $d_{nt}$ is the binary indicator of treatment (i.e., $d_{nt} = 1$ if unit $n$ treated in time period $t$, otherwise $d_{nt} = 0$). Finally, $u_{nt}$ is the residual.

From your quote, Goodman-Bacon (they are the same person) suggests an event-study design as a possible alternative to the TWFE when there is staggered treatment. This is confirmed on the FAQ from Goodman-Bacon's website (see the "2. “Should I do an event-study?” where Goodman-Bacon discusses the benefits of an event-study design vs. TWFE).

An event-study design can be written down as follows. Let $s_n$ denote the first time period where unit $n$ is treated (i.e., the smallest $t$ such that $d_{nt} = 1$). We require that once $d_{nt} = 1$ it can never revert back to $0$. The equation to estimate is then something like

$$ y_{nt} = \alpha_n + \mu_t + \sum_{h=-\infty}^{\infty} \mathbb{I}(s_n = t+h) \beta_h + u_{nt} $$ so that $\beta_h$ is the "effect" of treatment $h$ periods after treatment (negative $h$ indicate before treatment), provided we fix $\beta_{-1} = 0$ for identification. Thus,

1. $\beta_0$ is the immediate effect of treatment
2. $\beta_1$ is the effect of treatment 1 period later
3. and so on...

In practice, you can estimate these coefficients by including sufficiently many leads and lags of the treatment indicator $d_{nt}$ in your formula/design matrix. Note that you don't actually estimate infinitely many $\beta_h$ values; how many you need depends on how soon the first unit is treated and how late the last unit is treated.

This paper by Sun and Abraham is cited by Goodman-Bacon and discusses the event study design in much greater depth.

Suppose the original TWFE equation was something like the following: $$ y_{nt} = \alpha_n + \mu_t + d_{nt}\beta + u_{nt} $$ where $n$ is the index for units (e.g., individuals, firms) and $t$ indexes time. $\alpha_n$ is the usual unit fixed effect, while $\mu_t$ is an intercept for the time period (often called a "time fixed effect," but some people don't like that term). $d_{nt}$ is the binary indicator of treatment (i.e., $d_{nt} = 1$ if unit $n$ treated in time period $t$, otherwise $d_{nt} = 0$). Finally, $u_{nt}$ is the residual.

From your quote, Goodman-Bacon (they are the same person) suggests an event-study design as a possible alternative to the TWFE when there is staggered treatment. This is confirmed on the FAQ from Goodman-Bacon's website (see the "2. “Should I do an event-study?” where Goodman-Bacon discusses the benefits of an event-study design vs. TWFE).

An event-study design can be written down as follows. Let $s_n$ denote the first time period where unit $n$ is treated (i.e., the smallest $t$ such that $d_{nt} = 1$). We require that once $d_{nt} = 1$ it can never revert back to $0$. The equation to estimate is then something like

$$ y_{nt} = \alpha_n + \mu_t + \sum_{h=-\infty}^{\infty} \mathbb{I}(t = s_n + h) \beta_h + u_{nt} $$ so that $\beta_h$ is the "effect" of treatment $h$ periods after treatment (negative $h$ indicate before treatment). Thus,

1. $\beta_0$ is the immediate effect of treatment
2. $\beta_1$ is the effect of treatment 1 period later
3. and so on...

Unlike TWFE, this allows the effect of treatment to vary over time (the issue bolded in the question).

In practice, you can estimate these coefficients by including sufficiently many leads and lags of the treatment indicator $d_{nt}$ in your formula/design matrix. Note that you don't actually estimate infinitely many $\beta_h$ values; how many you need depends on how soon the first unit is treated and how late the last unit is treated.

This paper by Sun and Abraham is cited by Goodman-Bacon and discusses the event study design in much greater depth.