How many 5-letter words from ABRACADABRA

Let's only consider the case when there is only one pair of repeated letters, such as VVXYZ.

You're correct to think that VVXYZ has $5!$ configurations with $2!$ repeated letters, so the number of ways of arranging VVXYZ is $\frac{5!}{2!}$.

But we're not just arranging VVXYZ in this example. There are many ways to get one pair of repeated letters, such as AABRC or BBRCD or RRCDA...

So there are 3 possibilities for the double-letter pair (either AA or BB or RR) - that's why you need to multiply by 3.

Then, for each double-letter pair, there are four possibilities (technically, 4C3) for the remaining letters (such as AABRC, AABRD, AABCD, AARCD). That's why you also need to multiply by 4.