Protection encompasses policies which restrict international trade. A Free Trader is someone who advocates for free trade, or unrestricted international trade, and would thus be opposed to protection. Hannay was, prior to the story, living in Rhodesia, a colony. With that background taken care of we can better understand the quote.
You’re a Free Trader and can tell our people what a wash-out Protection is in the Colonies.
Sir Harry calls Hannay a "Free Trader". Harry believes that Hannay is, politically, on the side of free trade. Because of this, Harry wants Hannay to "tell our people what a wash-out Protection is in the Colonies". ("You’re a Free Trader and can tell" can be better understood as "You’re a Free Trader, and that means you can tell")
There are a lot of possible meanings for the idiom "wash-out", but the one that makes the most sense here is "a failure or disappointment" (meaning 11). Harry wants Hannay to talk about what a failure Protection is; rephrased, how Protection (the economic/political policy) is such a failure.
Harry adds the qualifier "in the Colonies" at the end. Specifically, Harry wants Hannay to talk about how the policies of Protection are a failure within the colony of Rhodesia - how the policies are not achieving their goals (failing), or how they are having disastrous side effects. He believes that protectionist policies are failing, and that Hannay, recently from the colonies, would be equipped to explain/understand their effects in further detail.
All put together, here's a version of your quote, reworded for clarity.
Now you must be a good person and help me, because I need it. You are a supporter of free trade, and thus can well explain how protectionist policies are a failure in the colonies. All of you people are great talkers, good at convincing—I wish to Heaven I had such skills. If you do this I'll be in your debt forever.
This part of the answer isn't essential to understanding the quote - the meaning doesn't change if what Sir Harry thinks doesn't reflect economic reality - but I thought it would be helpful for establishing background.
Frustratingly, I'm having trouble finding specific references on whether protectionism was actually a problem in Rhodesia around the time the story is set (mid-1914). This is partially because I'm not sure where in Rhodesia Hannay was living in, and there are a bunch of places with similar names.
As @mikado mentions in a comment, "Imperial Preference" was a system which restricted colonial trade out of the British Empire, in order to benefit the mother country. Rhodesia was run by the British South Africa Company. They (predictably, as they were for-profit) tried to maximize their economic gains from the region.
The economy of Rhodesia (both the later-Zimbabwe and the later-Zambia bits) was heavily based on metal-mining, with some farming, mostly to support the miners. It's likely (and here again we run into my frustration with lack of sources) that Imperial Preference led to the precious metals being routed only to places in the British Empire, to keep their value in-house.
That's definitely something that happened during the 1932 Imperial Economic Conference, which led to a series of bilateral (two-way) trade agreements. These used tariffs to preference intra-Empire trade. This was explicitly about Imperial Preference, but it's unfortunately out of the time period of the novel.
An inability to trade the valuable products of Rhodesian labor freely, due to protectionist policies, may have led to a perception that the economy was not doing as well as it could be if free trade was encouraged.
It's also possible that by saying "in the Colonies" instead of a specific colony, Harry was referring to the colonies in general, in which case Imperial Preference was quite in effect. Whether the policies were failing would depend on the place, and probably be a matter of opinion.