If I share 50% of my DNA with both parents, why do I look so much more like one of them?
You got 50% of your DNA from each parent, but appearance is decided by a small subset of variants, and the random draw can hand you most of the looks-related ones from a single side.
Yes, you inherited half your DNA from each parent. But 'looks like' is driven by a few hundred variants that influence facial features, skin tone, hair, and eye shape. Which half of each parent's DNA you received is random, and that draw can give you most of the appearance-relevant variants from one side. So you can be 50/50 by total DNA and 80/20 by visible-trait DNA. That is not a contradiction, it is exactly what the maths predicts will sometimes happen.
There is also dominance. If your mother passed a dominant variant for, say, a strong jawline and your father a recessive one, you show your mother's jawline even though you carry both. Your own children could still inherit the recessive version and resemble your father's side.
To add to Arjun's answer: parents often notice a child looks like one of them for a few years, then seems to switch. That is real too. The genes shaping facial structure express at different rates as the face grows, so the same child at 3, 13, and 30 can convincingly resemble a different parent in each photo.