Is there a way to gauge the contribution of each unit in a fixed effects regression? Suppose we run a regression with one key independent variable and also include group fixed effects.
Is there a way to find out which groups contribute towards the observed results?

Software will help you extract and visualize the fixed effects so you can get a handle on their contribution to the model fit. In R, fixef() is a helper function that can be found in numerous packages. It will return a list of the fixed coefficients in whatever dimension that's most appealing to you.
Let's look at a very quick example. After loading the Produc dataset in R, I will regress the unemployment rate on state output, then adjust for state and year fixed effects. I am a bit partial to the feols() funciton in the fixest package, so let's go ahead and estimate the model and then extract the fixed effects.
# library(fixest)

mod <- feols(unemp ~ gsp | state + year, data = Produc)  # Estimate
fe <- fixef(mod)                                         # Extract
summary(fe)                                              # Summarize

Fixed_effects coefficients
                        state year
Number of fixed-effects    48   17
Number of references        0    1
Mean                     5.95 1.97
Standard-deviation       2.19 1.49

COEFFICIENTS:
  state: ALABAMA ARIZONA ARKANSAS CALIFORNIA COLORADO                 
           6.904   5.433    5.711      13.49    4.366 ... 43 remaining
-----
  year: 1970   1971   1972     1973   1974                 
           0 0.8031 0.3987 -0.07369 0.5803 ... 12 remaining

The output offers a neat summary of the overall mean and standard deviation. We can see that most of the heterogeneity is across states. By construction, the elements of the first fixed effect dimension (i.e., state) will not be used at references. In the second dimension, however, we see the year 1970 was set as a common reference for us.
Once we store the results, the plot() function centers, sorts, and visualizes the "most notable" (i.e., highest and lowest) of the fixed effects.
plot(fe)


To be honest, the plot was a little cluttered so I used state abbreviations and omitted some of the state effects. I only did this for ease of viewing. Note California's unemployment rate relative to the national average. I should note that the interpretive value of these plots become a bit murky in settings with unbalanced fixed effects across multiple dimensions. Results will also differ in the presence of the state effects alone. I recommend this more so as a visualization tool.
