PHB Ranger: on the favored terrain and traveling for 1+ hours, the party's movement is not slowed
Choose one type of favored terrain: arctic, coast, desert, forest, grassland, mountain, swamp, or the Underdark. [...]
While traveling for an hour or more in your favored terrain, you gain
the following benefits:
Difficult terrain doesn’t slow your group’s travel.
You must choose only one favored terrain
Note that you must, at character creation choose one of the listed terrains as the favored one. And all the benefits apply only to it.
For that one terrain, the party's movement is not slowed
But if they choose swamp and the party is traveling with the ranger in a swamp for more than an hour then yes, their movement will not be slowed down by it.
The narrative reason for this being that the Ranger, being an expert in navigating such terrain, is able to pick safe and quick paths through what would normally be difficult terrain.
Remember that this is one of the things that Rangers are supposed to be good at
If your ranger chose, as part of his class, to be good at traveling across a certain type of terrain then you should let them be good at that. As a DM, allowing the players to feel good at things and powerful is one way to ensure that everybody is having fun.
Unearthed Arcana Revised Ranger: works similarly, but on all terrains
So if your player is using the UA Revised Ranger, then yes, they get the benefit of ignoring all difficult terrain no matter the source if traveling for more than one hour.
UA material is not finalized material and thus even more open to DM tweaking than other parts of the game.
There is nothing inherently overpowered about the ranger being able to ignore all types of difficult terrain unless overland travel and time management are meant to be very important mechanics in your game.
Note that Ranger is widely considered to be one of the most underpowered classes and the buffing of this ability was likely one way they were attempting to put the Ranger on equal footing with other classes. Again, traversing terrain is a mechanic that rangers are designed to be great at. So be careful if you want to remove this benefit from the game for your ranger.
Also, finding a way to narratively explain the character's power is a better option than removing the power because it does not make sense narratively I have found. If this is a concern for you, try that before removing or editing.