And it is a skill. I also have a highly-active internal thoughtstream, and it can cause me significant stress if I don't control it. In my case, it isn't a verbal dialogue, but rather the thoughts that swim around one level below. Busy nonetheless.
The only thing that works for me is to not fight it. If I attempt to resist the flow, it only amplifies the pressure, akin to pressing your thumb over the garden hose. Much of the stress of these thoughts comes not from the thoughts themselves, but the urge that feels like you must hold and analyze them all before moving on.
Rather, concentrate on letting it pass by, acknowledging it as it flows downstream, without reaching out to hold on. Visualizing yourself sitting calmly near a rapidly flowing creek. Imagine the strength of the flow slowing over time. Let go of the worry of the uncompleted thought- tell yourself that if it is truly worth considering, it will flow back around again in time. Smile to yourself as the thoughts wash through you, appreciating their fleeting nature. After a short amount of time, minutes at most, you will find the strength of the torrent ebbing. It will not cease, but the trickle that remains is much easier to cope with. Ten minutes of this, and you will find yourself mentally refreshed, and ready to get on with things.
This, of course, is not a cure-all, and if you expect it to work every time, you will be disappointed. But it does become easier with practice, and in my case, at least, it genuinely helps.
Others do what you're describing, let the thoughts come but see them as just passing clouds in the sky. Eventually you distance yourself from that stream of consciousness and can focus just on your breath.
The two are not mutually exclusive either.