The light bulb died. I haven’t come across this type before and honestly I am also unsure how to take it out. It seems like I might need to take out the plate.
There is probably a spring loaded contact on one end. Try moving the bulb gently both ways and see if moves. If so, move it far enough to remove it via one of the slots on either end of the bulb holder. You def don't need to remove the plate.
It looks like a halogen bulb, if so, when replacing it, don't touch the new one with bare hands, use a paper towel or tissue. Oils from skin will shorten bulb life. (this might be an old wives tale, but it's one I follow and it can't hurt!)
The socket is called
R7s. These halogen light bulbs have lengths of 78 mm, 118 mm, 189 mm or 254 mm. From the picture it seems to be an 118 mm, which is most common. (Measure or read the imprint)
You find the wattage on the side of the bulb or sometimes in the ceramic socket. It must not exceed
More important, the PE (Protective Earth) is badly connected.
- the copper is corroded and the twisted wires might have bad contact. Likely the second wire is needed to protect another device.
- there is no heat protection for the PE wire
- the screw is corroded too and there is no good contact to the reflector
It is the older type of halogen fitting, with a spring loaded pin at each end. Despite a comment, it's usually fitted outdoors - it's often a high wattage, and will get hot, so outdoors is sensible.
Be sure that the power to it is off, as it may not have blown, but the contacts may have corroded slightly. In any case, wear a fabric glove to push it to one side, then the other. It's quite stiff, and the protection is for several purposes. If it did come on, you'd burn yourself - they get very hot. That's why the power needs to be off. The replacement may suddenly be activated, also making you fall off the ladder - been there, done that! But also, grease/oil on skin will shorten the bulb's life tremendously. All that apart, it may break - cutting you.
There are various wattages available - along with at least two different sizes, physically. As in the comments, a complete replacement with an LED, although not so cheap, will save money in the long run. Having said that, these are used mainly in security lights, which are rarely on for more than a couple of minutes - usually triggered through heat/motion. (That's why I say make sure power is off - depending on settings, it could be triggered whenever!)
Lastly, it's very easy to locate one metal end on its pin, while the other ends up on the insulation - and it just won't light...