Being skinny does not automatically make someone fit, nor does it mean that someone is healthy.
"Physical fitness is one’s ability to execute daily activities with optimal performance, endurance, and strength with the management of disease, fatigue, and stress and reduced sedentary behaviour" (https://link.springer.com/referenceworkentry/10.1007/978-1-4419-1005-9_1167
To become more fit, you'll need to improve your stamina and strength, in terms of your muscles and your cardiorespiratory system. Essentially, to do this, you start small and gradually build up, at a level where you're pushing yourself a bit, but not so much that you're completely exhausted all the time.
e.g. for running: try running for a minute, walk for a minute, and repeat 5-10 times. As this gets easier, increase the length of time you are running, and decrease the length of time you are walking. You could also do similar with lamp posts (e.g. run the distance between 2 lamp posts, then walk, etc.), and also mix up the speed, hills, etc. Make sure you have proper running shoes, have warmed up suitably, are eating and drinking enough throughout the day, and don't go too hard too fast – build up gradually, and have rest days, otherwise you'll end up injured. Also mix things up with walking (hills/stairs in particular get your heart pumping, but also just a nice walk along the canal or something is always lovely), cycling, swimming etc., whatever makes you happy and feel good.
Might be a good idea to speak to a GP regarding your weight, just to check if everything's ok.
I personally have found that I am at my strongest and fittest now, since gaining a decent amount of weight. I'm also happier, not cold all the time (and less bothered by the cold), and overall healthier.
Eat more (so that energy intake exceeds energy output), and be mindful of how much exercise you're doing (i.e. if you increase your exercise, make sure you eat more) – this is at a general sense, so don't restrict your intake on days you move less. Your body needs fuel.
Anyways, speak to a GP, try to gain some weight, and see where things go from there