This is a standard regression equation, in the context of an econometric approach. The authors postulate a theoretical relationship between the variable "Change in Cash Holdings", which is treated as the "dependent variable", while "Cash Flow" and the rest are the explanatory variables/regressors.
The "factors in front of them" are the regression coefficients, which are unknown, and which are to be estimated through a regression estimation approach, like Ordinary Least Squares or any other technique deemed appropriate, using a sample of observations, which could be either "cross-sectional" (one balance sheet for each company, many companies), "time series" (many consecutive balance sheets for one company), or "panel data" (many consecutive balance sheets for many companies).
Each coefficient represents the partial marginal effect of a change in the regressor on the dependent variable, holding the other regressors constant.
There is also a "stand alone" coefficient, $\alpha_0$: this is the "constant term of the regression". Attached to it there is an implied "explanatory variable" that takes the value $1$ throughout the sample. It captures the average value of the dependent variable after the average effect of the regressors has been subtracted. The $\varepsilon$ at the end is the regression error, and it is unknown -an estimation of it will be given by the series of residuals obtain as the difference between the estimated value of the dependent variable and the actual one.
In Table 2 p. 1794 "The Baseline Regression Model" in the paper contains the numerical estimates the authors obtained for the $\alpha$'s for the specific regression equation, based on a panel data set, it appears, and for a number of variants.
According to the authors, immediately below eq. $(8)$ they identify the "Cash Flow Sensitivity of Cash" with the unknown coefficient $\alpha_1$. As already mentioned in the question (and as can be seen in Table 2), they find that the estimated value of $\alpha_1$ is $0.05-0.06$.
As described immediately above eq. $(8)$, "Cash Flow" here is essentially Operating Profits plus Interest Costs minus Dividends. Then the interpretation is that "on average, for every dollar increase in thus defined "Cash Flow", the Cash Holdings of a company increase by 5-6 cents" (the authors use, perhaps misleadingly, the verb "save" instead of "increase". "Hoard" would also be accurate).
"Applying" a regression equation to a specific firm, means, use time-series data on that firm, run the regression as specified and obtain firm-specific estimates for the coefficients. You essentially "accept" the theoretical validity of the regression specification, and you attempt to find the values of the coefficient pertinent to a specific firm (remember, the obtained estimates by the authors are averages over data from thousands of firms). You can then compare these estimates with the estimates obtained by the authors and see whether the firm under examination is close to average or not.