What does it mean for there to be more variance within groups than between groups in IQ?
In the video he is discussing some work of Charles Murray (not sure how familiar you are with it). In any case, the main point of your confusion comes from the difference between presence and size. Dr. Harris admits that there are differential IQ values between the races: there is (presence) a difference. The immediate follow up, though, is that the size of that difference pales in comparison to the variability with the group, indicating that no practical conclusions can be made.

For a silly example (and not leave any Earth groups out), consider Martians and Plutons (poor Pluto...): Martians may have a slightly higher IQ than Plutons, but the variability within each group is much higher than the between group difference. Dr. Harris' whole point is that, upon meeting a random Pluton and a random Martian, you practically have no idea which of them will be smarter, even though we know that Martians tend to be smarter. A pretty picture illustrates the two groups below.

I hope this helps! Take home message from the video is that even though there are small differences between races in terms of IQ, they should not play a role in someone's judgement of another person due to the immensely higher variability of IQ within any given race (there are some super smart Plutons and some super dumb Plutons, just like there are some super smart Martians and some super dumb Martians: you won't know which you are speaking to until you actually talk to them).



Graph coded in R:

boxplot(cbind(Plutons = rnorm(1000, mean=100, sd=15), Martians = rnorm(1000, mean=102, sd=15)))