The intuition you need is something like this–

1. an understanding of the economic prosperity of typical travellers in the cities involved in your routing,
2. an understanding of the "politics" of the carriers involved (beyond simple economics), and
3. a basic grasp of the mechanics of airline pricing.

They are not totally separate but let us look at some examples.

Economic prosperity.

Stopping in London is expensive.

Typical example, you wish to fly from New York to Copenhagen.

Let's say you need no flexibility, you are doing a return trip, you can stay a Saturday night, you can give one month's notice, and you are willing to tolerate economy class: you want the deep discount fare. Right now BA/AA/and their partners will sell you a return trip for 692 USD in early September.

This is actually a flight via London. You want to visit London (hypothetically). How much can it be to stop there?

[By a "stop" I mean a stop of over 24 hours. Unless otherwise specified if the time between flights is less than 24 hours on an international ticket, the flights are considered connecting and stopover restrictions do not apply.]

Investigating the fare rules reveals it is a throughfare on NYC-CPH; that is to say, the outbound part of the pricing unit is not composed of two separate fares. Can this fare cover us if we make a stopover (>24 hours) in London? No. Category 8 (Stopovers) says—

Category 8: Stopover restrictions
  NO STOPOVERS PERMITTED.

No, so we will have to reprice everything. Repricing with a single 26 hour stop in London and the cheapest price (covering almost exactly the same flights) is now 1445 USD. Why?

Travellers between London and New York have a high willingness to pay. You know the sheer quantity of business transacted between those two cities means there are many weekly commuters on those flights.

Travellers on New York to Copenhagen are a completely different market. The prices reflect the market, not the route. And the fares are carefully calibrated by people using large computational models to try to separate the two markets so they can be separately priced.

If you want a free stop in London on a BA fare that happens to pass through LON, forget it. This is British Airway's base and the city is stuffed full of people willing to pay a lot of money to travel on their planes. BA will protect the margin carefully. This also applies to their sister company Iberia and their oneworld joint venture partners American Airlines and Finnair.

Stopping in Reykjavik is cheap.

As an alternative suppose you decided to fly Air Icelandic from Anchorage to London via Reykjavik. Reykjavik is not a major centre of commerce and there is little marginal revenue for Air Icelandic to protect from spoilage. Therefore stopovers are free on the fares sold by Air Icelandic, although the tax/fee and availability computation may affect the final price.

Indeed Reykjavik is an inconvenient place to stop for most travellers so the airline tries to sweeten the deal with flexibility on the stopovers.

You can see that begins to shine through in the fare restrictions:

Category 8: Stopover restrictions
  1 STOPOVER PERMITTED IN EACH DIRECTION IN REK.
     A STOPOVER MAY NOT EXCEED 177 HOURS.
         NOTE -
          STOPOVER IS 7 DAYS/177 HOURS IS TO ALLOW FOR
          FI FLIGHT SCHEDULE
  AND - 1 STOPOVER PERMITTED IN EACH DIRECTION ONLY AT THE
            GATEWAY IN THE UNITED STATES/CANADA AT USD 40.00
            EACH.
           A STOPOVER MAY NOT EXCEED 177 HOURS.
         NOTE -
          STOPOVER IS 7 DAYS/177 HOURS IS TO ALLOW FOR
          FI FLIGHT SCHEDULE

[REK is the city code for Reykjavik covering both airports there, as NYC is for New York or LON is for London. FI is the airline code for Air Icelandic.]

But then why are "poor" cities so expensive?

Some far flung cities seem like they are massively overpriced: how could the average local resident of this place* afford the prices from there to London? But then you look at the profile of the customers on the flight: NGOs, western charities, oil company representatives, adventure tourists. Some places don't attract the kind of demand required to bring the prices down.

*(I don't want to insult anyone so let's leave the city blank.)

Airline "politics" (or political economics)

Emirates wants you to visit Dubai.

Let's look at another example: Madrid to Osaka via Dubai on Emirates. Same circumstances/dates as before. Cheapest fare in September is presently 729 Euros return.

Can we do a stopover without repricing the fare?

Category 8: Stopover restrictions
  8 STOPOVERS PERMITTED ON THE PRICING UNIT
     LIMITED TO 6 FREE AND 2 AT EUR 75.00 EACH.
     CHILD/INFANT DISCOUNTS APPLY.
         NOTE -
          STOPOVERS ARE PERMITTED ONLY AT THE CITIES
          MENTIONED IN THE ROUTING.

Yes. Actually because the airport fees are recalculated on a stopover it adds about 40 EUR to the final price. (Why are there eight permitted when the route could not possibly support that many? Well the fare rules are generated by macros to cover many scenarios.)

But Dubai is an increasingly important commercial centre, it is no longer a desert backwater with a runway. Why is EK willing to allow stopovers here, for free, below the market price? I believe EK thinks that the more people spend time in Dubai, the more business will be attracted to the city and their future prospects will grow accordingly. Alternatively the Dubai Emirate may sponsor it. But it is only a guess.

Qatar keeps tight control of policy

Let us look at one of their major competitors, Qatar, who do a similar route under similar circumstances: MAD-DOH-KIX and back, 709 EUR without the stop. Taking the stop adds about 100 EUR. So not enormous but you can see, it isn't free.

Now I know from a friend at Qatar Airways that their revenue management team are seriously annoyed by hidden city ticketing. This is when you buy one fare on AAA-BBB-CCC but simply exit the airport at B, paying a lower price than the airline intended.

And indeed you can see that irritation is entirely present in the fare rules:

Category 8: Stopover restrictions
  5 STOPOVERS PERMITTED ON THE PRICING UNIT
     LIMITED TO 1 FREE AND 4 AT USD 100.00 EACH.
       1 FREE IN EUROPE/AFRICA
       1 FREE IN AREA 3/AREA 1
       1 IN DOH EITHER OUT/IN AT USD 100.00
       4 IN AREA 3/AREA 1/EUROPE/AFRICA AT USD 100.00 EACH.
     A STOPOVER MAY NOT EXCEED 7 DAYS.
         NOTE -
          PURCHASE OF AN OFFICIAL QR HOLIDAY STOPOVER
          PACKAGE WILL SUPERSEDE THE ABOVE RULES STOPOVER
          FEES AND RESTRICTIONS. REFER TO QR OFFICES AND QR
          HOLIDAYS FOR MORE INFORMATION ABOUT THE OFFICIAL
          STOPOVER PACKAGE AND BOOKING PROCESS.
          PURCHASE OF ONEWORLD MIDDLE EAST PASS IN
          CONJUCTION WITH INTERNATIONAL QR TICKET WILL
          SUPERSEDE THE ABOVE RULE STOPOVER FEE AND
          RESTRICTIONS. REFER TO INFORMATION ON WORLD
          MIDDLE EAST PASS. AGENT SHOULD CONTACT QR OFFICE
          AND PROVIDE PNR DETAILS AND TICKET DETAILS OF
          BOTH LONG HAUL TICKET AND ME PASS TICKET.SAME
          INFORMATION SHOULD BE FORWARDED BY QR OFFICE.
          --------------------------------------------------
          ALL STOPOVER FOR THE PURPOSE OF CONNECTION STPC
          MAY BE PROVIDED FOR ITINERARIES WHERE SHORTEST
          STOPOVER OPTION IS MORE THAN 8 HOURS AND LESS
          THAN 24 HORUS FOR FULL DETAILS CONTACT LOCAL QR
          OFFICE OR REFER TO GGAIRQRSTPC IN AMADEUS.
          -----------------------------------------------
          IF THE STOPOVER IN DOHA BETWEEN CONNECTING
          FLIGHTS IS LESS THAN 24 HOURS PLEASE NOTE THAT
          THE BAGS WILL ALWAYS BE CHECKED THROUGH TO THE
          FINAL DESTINATION.THE PASSENGERS SHOULD CARRY
          EVERYTHING THEY NEED IN DOHA IN A HAND BAG.
          -----------------------------------------------
          IN CASE THE PASSENGER NEEDS THE BAGS IN DOHA
          TWO SEPARATE TICKETS NEED TO BE BOUGHT.

No other airline bothers to include those final two paragraphs, but Qatar do because they want to make clear, you don't get to stop in Doha unless you buy the right fare.

Qatar also include the following warning to travel agents in the sales restrictions category,

   TICKETS MUST BE ISSUED BY ELECTRONIC TICKETING.
     NOTE -
      ALL BOOKINGS MUST ADHERE FULLY TO QR BOOKING
      POLICY AND BOOKING MUST ALWAYS BE DONE AS PER
      FULL ORIGIN AND DESTINATION AVAILABILITY.
      ANY VIOLATION WILL BE SUBJECT TO ADM AND ACCESS
      TO QR INVENTORY WILL BE CLOSED IN CASE OF
      PERSISTENT VIOLATIONS.

The reference to "origin and destination availability" is again a sort-of hidden city ticketing thing. ADM is an "agency debit memo", i.e., an invoice for payment of the fare your customer should have paid!

Mechanics of airline pricing

Some things to watch out for are as below.

1. On an international ticket, it's usually possible to arrange a long layover of under 24 hours without affecting the price (but you might not see this option unless you ask for it), and this applies if any part of the travel on the ticket is international. e.g., in a couple of weeks I have carefully calculated layovers of about 10-20 hours each in London, New York, Sao Paulo, Dallas, Austin and London again.
2. If you are pricing a multicity journey internationally, it is often cheaper to do the "middle" leg on a completely separate ticket, and buy the outer legs as an open jaw. This is because (typically) once you start including more than two international fare components, the very cheapest fares are disallowed and we jump into more expensive fare territory. Whereas it is usually possible to combine two deep discount return fares into an "open jaw" journey on a half-round-trip basis.
3. Domestic add-ons in small countries are usually free. For instance I used to live in Leeds in the north of England, the 30 minute hop from Leeds to Heathrow was always free on my longhaul fares. Sometimes it is possible to have a stopover at the "gateway" city (the first city before the international flight).
4. It is often cheaper to begin your trip in another country, even if the first flight is back to your own country. This is especially true with business class fares. The trip I mentioned before actually "begins" in Bergen, Norway, saving me 3500 GBP over the listed price from London for the cost of one flight+hotel to Bergen and the cost in time of a night in Bergen.
5. Unless you are flying to a place that experiences very predictable and very strong seasonal demand, don't rush to buy your flights. The price is usually flat until about six weeks before hand, in my experience; except that sales might happen between now and then. That said, if you see a great price for a trip you need, buy it instantly. It won't last long.
6. Often the airline allows you to take an indirect route for the same price (+ extra airport fees). The search engines don't show you this unless the direct flights are full, because they think you don't want to do this. But this way you can visit more interesting places and travel on new airlines. On a trip to Australia in September I have managed to route via Beijing, Hong Kong, and Tokyo instead of an hour in Singapore. The price went up by a few percent but mostly it was taxes.
7. I will add to this as I think of any more ...

Of course I will get many comments telling me why these "intuitions" are wrong based on cases here and there. There are exceptions to everything I've written but they are my general observations.