Clarification on the regression coefficients
Cochrane (Asset Pricing, rev. edition, 2005) states (p. 247):
It it easier to do this in a more standard setup, with left-hand variable $y$ and right-hand variable $x$. Consider a regression
$$y_{it} = \beta´x_{it} + \epsilon_{it}$$
$$i = 1,2,..,N$$
$$t = 1,2,...,T$$
[...] In an expected return-beta asset pricing model, the $x_{it}$ stands for the $\beta_i$ and $\beta$ stands for $\lambda$.
Background
The Fama/MacBeth procedure is used to estimate consistent standard errors in the presence of cross-sectional correlation.
Fama-MacBeth (1973) - First step
The first step is a time series regression to get your right-hand variable $x_{it}$, i.e. the beta coefficients. As you are already aware of the technical details, let me just refer you to these answers [1], [2], [3] with further details on this step.
Fama-MacBeth (1973) - Second step
The gamma coefficients (here: $\lambda´_t$) are estimates for the risk-premium of your risk-factors $\beta´_t$. What does this mean? We apply a cross-sectional regression at each point of time $t$. If there is a (linear) relationship between your risk factors $\beta´_t$ and stock returns in period $t$, we would obtain a well-measured (i.e. statistical significant) positive factor risk-premium at $t$. The economic interpretation of $\lambda´_t$ is how much the expected stock return would rise, if this stocks risk-factor increases one unit.
We get estimates for the risk-premia $\lambda´_t$ at each point of time $t$. Due to limited computational power (and statistical methodologies) in 1973, we simply use the variation in $\lambda´_t$ over time to deduce its variation across samples.
You may look at this excellent answer on the technical details of this second step.
Fama-French three factor model
Your stated regression gives you the factor-loadings of a certain stock or portfolio. You may use these coefficients e.g. to calculate the expected return of this stock. However, the factor-returns are based on certain investment strategies (SMB/HML). As stated here,
you cannot interpret the average return for the factor as the risk premium.
but this needs further clarification, which follows now.
Conclusion
You may be confused by the term risk premium. The Fama/French factor time-series SMB or HML are indeed risk premiums (like the market-risk premium), but not in terms of the Fama/MacBeth procedure.
What Fama/French within their Three-factor model do, is to construct portfolios which follows certain investment strategies. These return series are risk-premia, because it measures how much a stock`s return should increase, if its beta for this factor increases one unit. We have strong empirical evidence, that these risk-factors drive stock returns.
Fama/MacBeth however start with risk-factors (like market-beta) and test, if there is any observable market-premium for this risk-factor in the cross-section of stock returns. If we would not see any significant and positive risk-premium, our risk-factor is not able to explain differences in the cross-section of stock returns.