Can you build a road through someone's settlement in Catan?
No. An opponent's settlement or city blocks your road. You can build up to it, but you can't start a new road on the far side of it.
The rule
Every new road must connect to one of your own roads, settlements or cities.
The 6th edition says it directly: you may not build a road starting on the other side of an opponent’s building. Your network stops at their settlement or city.
It works the other way too. If an opponent’s road reaches an empty intersection in the middle of your road, they can build a settlement there (following the distance rule). That splits your road into two shorter routes for Longest Road.
Example
Mia's road reaches a corner where Kai has a settlement. Mia can't build the next road segment out the other side of Kai's settlement. She has to go around.
Why players argue about it
The flip side surprises people: an opponent can build a settlement on an empty intersection along your road, which cuts it in two for Longest Road.