When you have an outcome $Y$, an endogenous explanatory variable $D$, a valid instrument $Z$, and controls $X$, then your two-stage least squares (2SLS) equation system would be
$$
\begin{align}
Y &= \alpha + \beta D + \gamma X + \epsilon \\[0.5em]
D &= a + \pi Z + \delta X + \eta
\end{align}
$$
where the second equation is the so-called "first stage" regression. Now rather than estimating these two equations step-by-step (as the name 2SLS suggests), the reduced form "reduces" the problem to a one equation estimation by substituting the first-stage into the second-stage:
$$
\begin{align}
Y &= \alpha + \beta (a + \pi Z + \delta X + \eta) + \gamma X + \epsilon \\[0.5em]
&= (\alpha + \beta a) + \beta \pi Z + (\gamma + \beta \delta)X + (\epsilon + \beta\eta)
\end{align}
$$
Note that this isn't really the official definition of "reduced form" but in my opinion it helps to think about it in this way. So this is what your researchers are doing. They regress $Y$ on $Z$ and obtain the parameter $\beta\pi$.
Now why do they do this? The first stage regression and the reduced form give you your 2SLS estimate because
$$
\beta_{2SLS} = \frac{Cov(Y,Z)}{Cov(D,Z)} = \frac{\beta\pi}{\pi} = \beta
$$
(after partialling out $X$) where $\beta\pi$ is our reduced form coefficient on $Z$, and $\pi$ is the first stage coefficient from the regression of $D$ on $Z$. Knowing the first stage and reduced form coefficients, you can tell
- the size of the 2SLS coefficient, $\beta_{2SLS}$
- the sign of the 2SLS coefficient
Personally I usually skip this exercise because in the end everyone runs their 2SLS regressions in order to also have the standard errors and confidence intervals for $\beta_{2SLS}$ and there you get your final result anyway.
Edit:
Also see Dimitriy V. Masterov's explanation of what reduced form means here. Also note that the above discussion is valid for one endogenous variable and one instrument. When you have multiple instruments, i.e. your endogenous variable is over-identified, $\beta_{2SLS}$ will be an optimally weighted GMM average and there is no way to directly get from the reduced form and first stage coefficients to the target parameter $\beta_{2SLS}$ because you would need to know the weights for each instrument (see these slides by Steve Pischke).
