In this age of data-oriented products, companies collect and store all sorts of data of their users. Big data is a term which refers to this profusion of data. But there is no use of just storing something unless it proves useful to the business of companies. What companies do is that they analyze this data to derive meaning and extract information out of it. For example, advertisement based companies like Google and Facebook extract user preferences and behavior to show users targeted ads which they would be interested in. E-commerce based companies like Amazon like to extract user preferences in order to recommend them products they are most likely to buy.
The term "huge value but very low density" means that the data contains a lot of useful hidden information, but it is not concentrated information. You cannot predict a user's choice by analyzing just two or three of his instances of his previous choices. You need to analyze a multitude of his previous choices, websites he visits, demographics, personal information and so on, to even get close to what he would like to buy.
Therefore, big data stores a lot of useful data for the company (huge value) but the information that is valuable is obtained only after analyzing a lot of data (low density).
To draw an analogy, you can think of big data as a diamond quarry. The quarry as a whole is of huge value as it contains a lot of diamonds, but the diamonds are not found all at the same place, they are scattered. So the density of diamonds in the soil is quite low.