What Tdol is referring to is more formally known as the debate on breadth versus depth of vocabulary. Breadth is the sheer number of words you know, depth refers to a number of related things such as different uses for any given word - different meanings, registers, collocations, morphology, etc.
There's a lot of ongoing debate and research suggesting which is more critical, with favor tipping from one to the other. The last formal training I had suggested the latter, although some articles lately seem to be swinging back towards breadth as the more critical.
In regards to reading, you need to know something like 8,000-9,000 word families to understand 95% of higher level English academic texts (and you need to know upwards of about 98% of the words to fully comprehend any given text). For lower level and daily functional English, studies show a vocabulary of around 2,000 word families will let you understand about 80% of what you read, but something like an additional 3,000
word families to get you up to 88%. After the first 2,000 word families, every additional thousand word families gains you an increasingly smaller percentile comprehension gain.
Note this all refers to word families, which average about 3 individual words per word family
You can find all kinds of information about this on the internet - research has been going on for decades.
Here are some articles and links to get you started reading.
Lengthy 21 page PDF
Study on Iranian EFL students (note additional PDFs from study available)
Yet another article
Article from Asian Social Science Magazine
For further exploration, here's the search results link
I used for these articles.