What is virtual work?

To understand virtual work you should understand virtual displacement first,

The wiki I have linked in first paragraph describes virtual displacement deeply. Usually we write $\delta$ or $d$ to say infinitesimal, now the question is infinitesimal what? As the OP wrote in question that $\sum _i{F}_i\cdot \delta r_i$ so when there's infinitesimal displacement there's infinitesimal work.

So if an object done infinitesimal displacement by force F then the work on that object is called virtual displacement.

Internal force is force done on that object. If force is huge then the displacement will be large but if force is infinitesimal (assuming force to be $0$) then displacement will be also infinitesimal that's what infinitesimal work.

If you want to read more about Virtual Work then visit Wiki page

The virtual work is called D'Alembert principle