This is a question that might be more suited to another subreddit like /r/economics, but there's at least one example from 20th century history that I think could be a useful introduction to this. As a disclaimer, this is by no means my area of academic specialty, but aviation is an interest of mine and I've looked at how infrastructure's developed around the world out of curiosity.

With luck, someone who's specialized in this will be able to answer in greater detail, but I hope this helps a little.

Boomerang money: I'll use Tanzania as a representative example, simply because that was how I first started learning about this, and because elements of Tanzania's development during the 20th century mirror North Korea's in strange ways . P.J. O'Rourke in Eat the Rich wrote a travel essay about the country in which this term figured prominently in a description of the economy's problems. In short, boomerang money is what happens when someone or something invests in a country but most of the profits don't remain there. For example, if you go on safari in Tanzania, odds are pretty good that:

• You'll arrive in a foreign-built plane owned by a foreign airline
• You'll stay in a hotel that was built by foreigners and is owned by an international corporation that's not headquartered in Tanzania, and:
• You'll eat food that's been imported because the country doesn't have large-scale agriculture and lacks the road and rail infrastructure to move mass quantities of food to urban areas efficiently anyway.

So "boomerang money" is money that comes to Tanzania, stops briefly, and then leaves. Foreigners love to come see the beautiful landscapes and animals, but they're generally spending money in places where the profits won't remain in the local economy.

That doesn't mean that Tanzania doesn't benefit from tourism, because it clearly does. By the World Travel and Tourism Council's recent estimates, it's ultimately responsible for about 13% of Tanzania's GDP (warning: PDF). Hotels and airports and restaurants employ locals, and tourism also benefits locally-owned shops and people who work as safari guides and park caretakers. It's just that a huge amount of money associated with tourism (and perhaps even most of it) doesn't remain in the country, which is problematic for Tanzania's long-term growth.

So why didn't Tanzania develop any of this stuff on its own? This, bizarrely enough, is where Tanzania actually bears an eerie resemblance to North Korea, which is my specialty and had an equally interesting trajectory in 20th century history. (And that's in addition to its flag carrier being the world's notorious "one-star" airline, but that's not what you're asking about here.)

Anyway! Tanzania had particularly bad problems with economic growth due to Julius Nyerere's policy of ujamaa between 1961 and 1985. Nyerere was the country's first president after Tanzania's independence from the U.K., and he implemented a very aggressive set of policies to transform the country into something approximating his vision of a more perfect African society. Some portions of ujamaa bear a strange resemblance to Kim il-Sung's policy of juche in North Korea, and both had the effect of isolating their nations from the international economy.

Now, ujamaa wasn't all bad, just as juche wasn't all bad. (Although I'm sort of using juche as a blanket term for Kim il-Sung's social policies in NK, which is not quite historically accurate. I can clarify later if you need.) Their respective emphases on education and the improvement of health statistics were actually very, very good. However, the forced collectivization of agriculture, nationalization of industries, and mania for "self-sufficiency" in both nations absolutely destroyed their economies; both countries became, and remain, heavily dependent on foreign aid. Nyerere and Kim il-Sung brooked no dissent and authorized their respective militaries to shoot or imprison the people who protested. Nyerere was never quite as cruel about this as Kim il-Sung was, but I'm sure the distinction didn't matter too much to the millions of Tanzanians who were forcibly uprooted and relocated (warning: PDF) during the "villagization" of their society.

Neither Tanzania nor North Korea studied or developed the kind of societal or physical infrastructure needed to support things like wide-scale tourism or international business. Neither of their leaders saw these as desirable for their societies at the time.

So what does this have to do with airports? The reason I bring all this up is because airports are a crucial part of a modern economy, but they're not something that just anyone can build, run, or maintain. When developing countries need to build or expand airports, they often find themselves buying the knowledge and productivity of developed countries in order to do it. Without better education and investment, nothing about that is likely to change.

So if we're going to look at developing countries around the globe and ask how airports benefit them, we have to ask:

• Who builds the runway and airport? A simple air strip that you can use for general aviation (e.g., the littler Cessnas or Pipers) is relatively easy to construct; you just need a flat, straight piece of land around 800 to 1,000 feet, and for light planes it doesn't even have to be paved. By contrast, a runway that can handle the bigger passenger jets needs to be built to certain specifications. Passenger jets that are the size and weight of a Boeing 737/Airbus 320 or larger (and on average, planes flying a transcontinental route are larger than these) need to land on well-maintained asphalt, concrete, or reinforced concrete. Even sleepy regional airports in the developed world often have at least one runway that's >9,000 feet. The runways need to be painted a certain way, need to have lights that can be reliably seen from a few miles away, and need to be built so they can be easily accessed by emergency vehicles for obvious reasons. The airport itself is slightly less specialized, but you still need an architect and builders who know what they're doing.

Often the locals don't have the expertise or experience to build airports to international standards (or, as is common in former colonial outposts, expand or improve on whatever infrastructure the colonial government left). For example, Julius Nyerere International Airport and Kilimanjaro International Airport contracted with a Dutch company to add additional terminals and expand their capacity, and that's a pretty common practice in the developing world. If your country doesn't have the materials or experience to improve its air infrastructure, you pay someone else to do it.

• Who's going to service the planes? Wrenching a passenger jet is a high-skill and high-stakes job; there are a lot of high-profile crashes that have happened because of improper maintenance even in developed countries. If an airline can't guarantee that skilled mechanics who are type-certified for whatever the airline's flying into an airport are going to be present, that means they have to fly some in (and often parts) if there's a problem. Any plane that's stuck on the ground is a plane that isn't generating revenue.
• Who's going to staff the airport? This is generally less specialized labor where you can pick up a lot of locals. People need to staff the airport shops, restaurants, banks, etc., and government workers will need to man the immigration and customs booths. However, revenues from this will naturally depend on how busy the airport is.
• Who's going to use the airport? A nation's flag carrier usually gets preferential treatment at its airports, whether it's in the form of getting a more desirable terminal or reduced landing fees. Airports sometimes make up for this by charging much higher fees to foreign carriers, which makes them more attractive as a revenue source. (However, the math behind who gets charged what and why is byzantine at best.)

So, as a TL:DR, I believe the ultimate answer to your question is a Yes, but ...

• Can airports bring jobs, tourism, and money to developing nations? Yes, but someone else might be cashing the final check ("Boomerang money"). Having an airport is certainly not going to hurt you, but it enables economic growth rather than creating it all on its own. People still need a reason to visit you in the first place.
• Can airports help bring technology and educated individuals to developing nations? Yes, but poor countries are quite likely to be reliant on the education and productivity of rich countries, because airports take specialized knowledge to build and maintain. As above, there still needs to be a reason to visit and invest in the country in the first place.