The way I think of it -- find the sibling pair you both descend from -- if your greatgrandparent and your relative's greatgrandparent were siblings then you both have to go up 3 generations to find the siblings you descend from ( you->parent->grandparent->greatgrandparent = 3 generations) --
in this case you are 3rd cousins.
But what if one of you has to go up 3 generations to get to the sibling pair, and the other has to go up 5 generations? That is not the same amount -- both of you have to go up three generations, then one has to go up another 2 more generations -- so you are 3rd cousins twice removed.
Thrice removed means - one of you has to go up 3 more generations than the other to get to the sibling pair you two are descended from.
[or - looking at it another way - one of you has to go up 3 more generations than the other to get to the common set of ancestors you are descended from -- I prefer to go to the sibling pair, because the number then totally matches for whether 1st cousin (up 1 generation to the sibling pair) or 3rd cousin (up 3 generations to the sibling pair) or 13th cousin (up 13 generations to the sibling pair)
-- if you go to the common ancestor, you have to subtract 1 -- common grandparent, that is up 2 generations, 2-1 = 1, so you are 1st cousins -- works out the same]