I read a line in the book "The Germanic vocabulary of Old English has not survived particularly well into the current period". This really confused me a lot. Isn't English a branch of Germanic languages? Why does it say in that way? What's the relationship between Old English and Germanic?
In the tree model of language evolution, languages change and evolve regularly over time, eventually becoming different enough from their ancestors to be called a new language. In this model, Old English descends from Anglo-Frisian, which descends from Proto-Germanic, so Old English is a Germanic language.
And for many purposes, this model works great! That's why it's so commonly taught: because it's a really useful model that's good at explaining things. And you're right, English is considered a Germanic language, because the vast majority of its grammar and most basic vocabulary comes from Germanic. But, like any model, the tree model makes some assumptions and simplifications that don't hold up perfectly in practice.
In reality, English didn't evolve in a vacuum—quite a lot of other languages have had an influence on it over the centuries. And in particular, a huge amount of our modern English vocabulary is descended from Romance rather than Germanic, imported when the French-speaking Normans conquered the British Isles around 1000 CE.
So when the book talks about "the Germanic vocabulary of Old English", it means the parts of the vocabulary that were inherited from Proto-Germanic, rather than borrowed from French or other places. There's still quite a lot of it—more than enough to justify calling English a Germanic language—but much less than in German, Dutch, or Icelandic.
P.S. If you're curious just how much of English comes from non-Germanic sources, check out Poul Anderson's Uncleftish Beholding—a basic introduction to atomic theory written using only native Germanic roots. So words like theory, borrowed from Norman French from Latin from Greek, have to be replaced by the pure Germanic beholding. It's a very entertaining read.