How can I book a flight plan found on Matrix Airfare Search?

I'm not aware of any online booking engine that lets you book arbitrary fare constructions, so in practice you've got four options:

1. Print out the fare construction and go to a travel agent in person. (This is what I do for personal travel.)
2. If you already have a personal relationship with an agent, or your company has a designated one, you can e-mail them and ask to book. ("Cold-emailing" random agencies with complex itineraries is unlikely to work, since they will assess it as a lot of work for low commission and uncertain reward.)
3. Call the airline and book over the phone. (Fiddly, because reading out and double-checking gibberish fare constructions is not much fun.)
4. For heavy users and experts only: Sign up to a GDS and become your own travel agent, so you can issue any ticket you like. This is not a straightforward process, setup fees are considerable and you're looking at at least $50/mo or so for a subscription, but it might be worthwhile if you fly several times a month. Here's a (partly outdated) Flyertalk thread about the topic.

We just created https://bookwithmatrix.com, a site that lets you simply copy and paste the itinerary from ITA Matrix and get links to the major OTAs (online travel agents) where you can make the booking. As jpatokal mentioned, for the most complex fares you may have to talk to a travel agent, but this looks at the fare construction and tries to build the itinerary for 95% of flights you can find.

Most likely where you are hitting issues is with the "sales city".

Your output from ITA shows a flight from South Africa to the US, but the price is displayed in US$. This means that within ITA you've changed the "Sales City" from it's default (which would have been CPT, your origin city) to somewhere in the US, otherwise it would have display the price in South African Rand.

Airlines frequently price fares differently depending on where you buy them - a trip bought in the US might cost more or less than the equivalent trip bought in the UK, Turkey or South Africa. In general, a travel agency in one location can not access the fares available for sale in a different location - and that includes online travel agencies.

When you book a flight directly from the airline they will often use the departure city as the "sales city", and that's what ITA defaults to if you don't change it. When that happens the price will be generated in local currency for that location, and using the pricing available for that country. By changing the sales city you can change the currency and the price, but you end up with something that can only be purchased in the country that you've set the sales city to.

Presuming you got the price you've quoted above with the sales city set to somewhere in the US, any US travel agent - including any US-based travel website - should be able to get you that price. As you haven't given dates I can't confirm that, however checking on Orbitz for a few dates in December I do see CPT-JFK available on Turkish Airways for ~US$790 - a few dollars more than you've got above, but in the same region.

According to this resource, you can use some of ITA's routing language with Hipmunk.

What I have done is this: (not just for Google Matrix but other flight search engines too)

1. Print out or write down all the travel agent secret codes™.

2. Telephone the airline, tell them how I found the flight and that I need more details, asking the which of the secret codes™ they need me to read out to them.

3. Write down the price they tell me, assuming it's not something special that's only available through certain agents etc. (I'm not claiming to understand much of this secret stuff.)

4. Armed with all this information, head to a physical travel agent or two.

5. If visiting more than one travel agent, make sure the last one has something like a "We guarantee to beat any price" offer. Flight Centre should have this policy in most countries they operate in.

6. Profit! (-:

Just call the airlines shown on the itinerary (if American, call American), and tell them the details of the flight. I booked mine easily, and I only paid an extra 25 bucks as booking charge. The price was as came up on the ITA search engine.