Most of the time when planets are detected we never actually see the planet. There are a couple rare exceptions, mostly in very young stellar systems when the planets are still really hot and haven't cooled off at all (our solar system left this phase billions of years ago, so its a short-lived period of time). But what about all the planets being found? They're found using one of two methods: the transit method and the radial velocity method. The transit method is the more conceptually simple of the two. Let's say you were an alien looking at our solar system and just staring at our sun. And lets say you were looking at 'edge-on,' so to speak, so that you could see all the planets pass in front of the sun. Well, when a planet passes in front of a star, it will block out some of the light, similar to an eclipse. Planets don't tend to be very bright compared to the sun, so you, as an observer far away, would see the sun get slightly dimmer. If you saw this dimming happen in a predictable pattern (for example, the Earth would pass in front of the sun every year) you could say that there must be some planet there. With a few more details (like how much the sun got dimmer, how long the dimming lasted) you can find out a few things about the planet that let you get headlines like "Earth-size planet found around nearby star" or something.
The other method is the radial velocity method. You know how when a fire truck moves by you the sound of the siren changes? Its because of something called the Doppler Effect. As an object moves towards you the sound has a higher pitch, or frequency, than normal. When the object moves away from you, the opposite happens. The pitch is lower. The same thing happens with light. So if a celestial object (like a star) is moving towards us, the light increases in frequency, while if it moves away from us, the frequency decreses.
We can measure this increase or decrease in light even if stars are light years away from us because they're so bright, and because telescope are very, very large. So what makes the star move? Planets. We all know that planets orbit around the sun, but that's not entirely true. The Earth goes around the sun, but the sun moves too. It moves a lot less because the sun is a lot bigger than the Earth but it responds to the Earth's motion nonetheless. We can observe stars light years away moving in response to planets orbiting them (and the stars responding), which is how we find about half of the planets we know about.
Okay, now what about these 'phantom' planets. This goes back to the first method of planet detection, known as the transit method. If there's just one planet going around another star, then that planet will orbit in a very predictable pattern. After you see the star get dimmer a couple times, you can measure the length of the dimming and the time in between the dimming and you can actually predict when the next transit event (planet passing in front of the star) will happen to the exact minute, sometimes even better. But what if there's another planet in the system? What happens then? Then your predictions will be off by a little bit, maybe a couple minutes here or there. So if you look for variations in the time between transits (aka transit timing variations) you can say something about whether or not there is a planet or a moon or something else in that system. You can constrain what it could be using simulations but the general idea is that you can use this to find planets.
So why are the planets 'phantom' planets? Usually they're planets we wouldn't be able to detect using the transit or radial velocity methods. This means that they tend to be smaller planets. Which is great for finding things of astrobiological interest, like Earth-like planets. The planet that was found recently is probably too big to be Earth-like, but astronomers are tracking all sorts of transiting systems looking for these 'phantoms'.
