This interaction between a fixed and a random factor allows for differences in behavior of the fixed factor among the random factors. Let's run that code on the data set, available in the MASS package in R. (I kept the short variable names provided in that copy of the data.)
BVmodel <- lmer(Y ~ V + (1|B/V), data=oats)
Linear mixed model fit by REML ['lmerMod']
Formula: Y ~ V + (1 | B/V)
REML criterion at convergence: 647.8
Min 1Q Median 3Q Max
-1.66511 -0.67545 -0.00126 0.74643 2.11366
Groups Name Variance Std.Dev.
V:B (Intercept) 19.26 4.389
B (Intercept) 214.48 14.645
Residual 524.28 22.897
Number of obs: 72, groups: V:B, 18; B, 6
Estimate Std. Error t value
(Intercept) 104.500 7.798 13.402
VMarvellous 5.292 7.079 0.748
VVictory -6.875 7.079 -0.971
Correlation of Fixed Effects:
VVictory -0.454 0.500
This gives fixed effects for two Varieties (expressed relative to the Intercept that represents the Yield of the "Golden rain" Variety) and sets of random effects for Blocks and for the Variety:Block interaction.
Now let's look at the random effects themselves; for this purpose the
ranef() function provides the clearest display, as it shows the random effects with respect to the immediately higher level in the hierarchy. (I omit some of the interaction effects, as they aren't needed to make the point.)
Notice that the 6 Block random effects (
$B) all add up to 0. These represent how Yield differs (randomly) among the Blocks. The 6 random effects representing the interaction between Block and the "Golden rain" Variety also sum to 0, as do those for the 6 interactions of each of the other 2 Varieties with Block (not shown).
The interaction between each Variety and Block allows the Yield of a Variety to differ (randomly) among Blocks, around the overall Yield for the Variety and the overall random effect for the Block. It doesn't alter the fixed effect associated with each Variety, whether shown as the Intercept for "Golden rain" or as differences from "Golden rain" for the other 2 varieties.
Different yields of the same Variety among different Blocks, even after accounting for between-Block random effects, might be expected in practice and might need to be accounted for. So rather than thinking about this as "remov[ing] variance related to differences between varieties," think about this as allowing for a source of variance within each variety that is related to potentially different behaviors among Blocks.