Klaus's answer has all the right explanations but I always find these things hard to follow without pictures, so here are some to clarify.
Note that in the pictures below we don't explicitly draw carbons or the hydrogens attached to them (we just mentally add enough hydrogens to make the number of bonds to each carbon equal 4).
First what is benzopyran?
The two numbers on the first structure tell us where the oxygen appears in the pyran ring and where the saturated carbon (with an extra hydrogen) appears. Both could be in different places. The second picture showns the numbering convention for each atom in the ring so we can keep track of where anything else is attached. The third picture shows what happens if we notionally add extra hydrogens on carbons 3 and 4 to give a fully saturated ring.
Now we add an OH (hydroxy) in certain positions:
The picture on the right shows how we number the parts of trihydroxy phenyl when it is attached to something else (it will be stuck on the 2-carbon on the benzopyran in the final compound).
Finally we have the epigallocatechin:
The "cis" indicates that the two hydrogens (mentally fill them in in positions 2 and 3) are on the same side of the benzopyran (the opposite side to the two wedge-like bonds on the OH and phenyl groups).
I hope that all makes some sort of sense. Note that none of the above bears any relation to how the molecules are made: this is all notional deconstruction for the purpose of explaining how the molecule is named.