The idea of $k$-means (and clustering in general for that matter) is to group points into clusters so that points within clusters are close together but points in different clusters are far apart. The within cluster variance is what's meant by the "conditional" variance and it's what $k$-means seeks to minimize (the mean of a sample is the least squares estimate of location).
Consider the following simulated data:
# cluster 1, normal(0, 1)
x1 = rnorm(50)
y1 = rnorm(50)
# cluster 2, normal(6, 1)
x2 = rnorm(50, mean=6)
y2 = rnorm(50, mean=6)
# plot the data
plot(c(x1, x2), c(y1, y2),
# estimate the conditional variances
# estimate the unconditional variances
Within clusters both $x$ and $y$ only have unit variance, but if we ignore the groups and calculate the variances across clusters they becomes much larger. In fact the increase in variance is really attributable to a shift in location, and that is what $k$-means tries to capture.