Well…quite easily, really.
Go to the countryside. Wait until night. Go outside. Look up.
Voila!
You have detected things which are light years away!
The closest star to us is Proxima Centauri at 4.24 light years away, but is invisible to the naked eye - blotted out by its far brighter cousins in the binary system of Alpha Centauri A and B.
These stars are 4.37 light years away.
This star system is the third brightest object in the night sky - the brightest (Sirius) is a measly 8ly away - but the second brightest (Canopus) is 310 light years away!
Depending on how you classify “visible", the furthest object that can be seen unaided is one of the stars in Casseiopeia - between 4,000 and 15,000 light years away.
So there you go, using the good old Mk I Eyeball, you can detect objects which are thousands of light years away!
Obviously, however, you can only see the huge stuff which is chucking out loads of light - which makes it really easy to see them!
Distance isn’t the big achievement, really.
The clever stuff that NASA does is spot tiny, dark things across that distance!
Exoplanet hunting is done through a variety of methods, such as:
Direct Imaging
Which you do through a whopping great telescope, or lots of little telescopes all wired up into an “interferometric array”, which (if you have a supercomputer) allows you to treat them as a single whopping great telescope.
This accounts for only a tiny number of detected exoplanet - but produces cool pictures!
Transit Detection
Rather than looking at pretty pictures, you simply take a graph of the light intensity emitted by a star.
Normally the power output is pretty much a constant, but when a planet passes in front of the star, the light emitted drops - like in the image.
By timing the gaps between each drop, you can work out how many planets there are, how far they are from the planet and so on.
This is by far the most common way of detecting exoplanets.
There are a few other ways of doing this - such as using Doppler shifts to work out how much the star “wobbles” due to the gravitational influence of orbiting bodies, and so on.
In the context of the furore about the recently discovered earth-like planets in the TRAPPIST system, it's also worth mentioning that 40 light years really isn't that far away!
It's practically in the same village as us - the neighbour a few doors down!
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