The first planet orbiting a star other than the Sun wasn't discovered until 1992 and since then the subject of exoplanets has gone from being something you argue about the existence of to a rapidly expanding field with new discoveries all the time. The experts who discussed exoplanets on In Our Time were Carolin Crawford (University of Cambridge), Don Pollacco (University of Warwick) and Suzanne Aigrain (University of Oxford).
One of the reasons it took so long to discover any extra-solar planets, despite people speculating about their existence for centuries, is that they are very hard to directly see. In fact I think they were saying that none of the known ones have actually been seen. Instead a variety of more indirect techniques are used to detect them, and these required both sophisticated technology & sophisticated knowledge of physics before they could be used. The technology needed to develop to a point where small differences in stars could be measured accurately and consistently over time. And the physics is required to both predict how a star without planets would behave and then to figure out what the differences from this prediction mean.
In the programme they ran through a variety of techniques used to detect planets. One of these is to look at the colour of the star's light and see if it's changing between blue-shifted & red-shifted over time. If the star has no planets then you won't detect that. When there's a planet orbiting the star it's not quite as straightforward as the planet circling the star, actually the star and the planet are both circling a point between them (that's a lot lot closer to the centre of the star than it is to the planet). So the star will seem to move back & forth relative to us observing it. This is biased towards detecting more massive planets, as they'll move the centre of gravity from the centre of the star more - so-called "hot jupiters" for instance, which are planets the mass of Jupiter that orbit close to their star.
Another method is to look for the changes in the star's light caused by the transiting of a planet across the face of the star. Obviously this is only possible to detect if the planet is orbiting in the right plane for us to see it. But if you have one transiting where we can detect it then you can detect the existence of other planets in that system by looking at the perturbations of the orbit of the one that transits. You can also detect things about a planet's atmosphere with this method. The changes in the light of the star can be used to tell you something about the size of the planet (in terms of diameter), and if you look at different wavelengths of light then you'll see varying diameters. This tells you when the atmosphere of the planet is thin enough to be transparent to that wavelength, and different gases absorb different wavelengths differently so you can figure out the gases that are present. Apparently you can even detect the presence of clouds using this technique.
Another method uses the phenomenon of gravitational lensing. If the light from a distant star passes by a closer to us star on it's way to the telescope then it will be bent by the gravity of the middle star. A planet orbiting that middle star will affect the lensing effect, and you can figure out things about the size & distance from the star by exactly how the lensing is affected.
If you use the first two methods together you can tell things about the density of the planet. Is it small & heavy? Is it big & fluffy? Or even small & fluffy? There seem to be a wide variety of planet types out there, not all of which are represented in our own solar system. There are also a wide variety of types of solar system out there - Pollaco pointed out that one reason there was argument about the reality of the first exoplanet discovered was because people were assuming that our own solar system was a good model for "all systems everywhere". It turns out it's not. The example they used in the programme was systems that have hot jupiters - the first exoplanet was one of these, and the very idea of a Jupiter type planet orbiting with a periodicity of only 4 days was almost unthinkable. They also talked about planetary systems detected around brown dwarves - stars which weren't quite massive enough to ignite at the end of the formation process. And planets around pulsars (again like the first ones detected) - and one of the experts (I think it was Crawford?) made a throwaway remark about how these are probably not the first planetary system for the star in question. Before a star becomes a pulsar it goes through a supernova explosion, which would probably destroy any original planets - the ones orbiting afterwards are probably secondary captures.
They also discussed looking for planets which might be habitable. Bragg asked if we are thinking about life like ourselves, or germs. The answer was (paraphrasing) "yes". At the moment no-one knows enough to know what we're looking for in terms of life on other planets, and at first we're obviously limited to things we know about life on Earth as a starting point for what to look for. So looking for rocky planets which are neither too big nor too small, that are in the right zone for liquid water. And other things about our own solar system might've been necessary - like the presence of Jupiter which draws away some of the comets that could bombarded Earth & wipe out all life. I think it was Aigrain who talked about other ways of detecting life - looking at what we can tell about the atmospheres of the planets. If there are very reactive gases present then they must be being made constantly - some of these we only know of biological processes that make them. So if one could detect such gases that'd be a sign of life.
It was a little bit of an odd In Our Time episode, because there was less of a sense of a narrative than they normally have. It felt like this is because the study of exoplanets is in its infancy - we're at a point where most of the work is data gathering. I mean in the sense that a lot of planets are being discovered and categorised, but as yet they're not classified and grouped into types. Nor are there overall theories about how solar systems in general work or were formed - it's now clear that the one we know isn't the only sort there can be, nor is it particularly typical of what we're detecting now.