There has to be something that says EF is parallel to GH for us to do what I think they're trying to do. Just from the screenshot, there's a picture that looks like they're parallel but no explicit statement they are. Let's assume it's out of the shot.
Given they're parallel, we get to apply the Triangle Proportionality Theorem aka the Intercept Theorem or as it's called here, The Side Splitter theorem. It's actually one of the first theorems of geometry, due to Thales, circa 600 BC. There are two Thales' Theorems; this is one of them.
These parallel lines make similar triangles and the scale factor is common among all the analogous ratios. In particular
[tex]\dfrac{EG}{GD} = \dfrac{ FH }{HD}[/tex]
[tex]GD = \dfrac{ EG\ \cdot \ HD}{FH} = \dfrac{(6)(14)}{8} = \dfrac{3(7)}{2}= \dfrac{21}{2}[/tex]
