Mechanically, ranger captures are essentially the same as they are in Vanilla PTA. They are a Ranged action. You roll 1d100 and subtract 25 as well as whatever modifiers the DM deems appropriate, and if you roll under the pokemon's capture rate, it is befriended.
Here is the main difference:
There is no Capture Styler. Rangers instead use a regular lasso for all ranger captures. This Lasso is good for 10 ranger captures before it breaks unless you have a feat which counteracts this. Ranger Lassos can be repaired in town for free at a Sherriff's office, or a Pokemon center. Ranger Lassos can be repaired to full usage in the wild by pokemon with the "Threaded" Capability on a roll of 16 or higher on 1d20. To create a new lasso after breaking an old one requires a length of rope at least 10 meters in length (Or 30 Feet, VERY approximately), or longer if desired. The amount in meters of rope you make this lasso, with one subtracted from it (To take the loop into account) is the maximum range of your Ranger capture.
Any Feat which mentions styler energy now refers to the state of your Ranger Lasso. It begins at a quality of 10 and is reduced by 1 for each capture you make.
The ranger companion acts as if it were one of the ranger's normal pokemon and gains exp and levels normally, however, they are still considered wild and are beholden to no one but their Ranger. They will treat every fight, even a friendly "No deaths allowed" duel, as if it were life-or-death and will go for the kill if they can manage it. For this reason, Ranger companions aren't allowed in friendly duels (unless you lie and say its tame, which could get you inter trouble), and are strictly banned from contests.