Altitude sickness is the reason people don't reach the top of Kilimanjaro, and in rare cases it's the reason people get seriously hurt. It's also manageable, as long as nobody hides symptoms.
What causes it
There's the same proportion of oxygen in the air at 5,000 metres as at sea level, but the pressure is about half, so every breath delivers roughly half as much. Your body adapts, given time. Altitude sickness is what happens when you go up faster than it can adapt. It has very little to do with fitness and a lot to do with speed, and nobody can predict in advance how they'll react.
Acute mountain sickness
The common form, and most climbers feel some of it. Headache is the signature symptom, usually with at least one of: nausea or loss of appetite, dizziness, unusual tiredness, or sleeping badly. Mild AMS is normal, and it's handled by not going higher until it settles, drinking, eating, and simple painkillers. It is not handled by pretending it isn't happening.
The two serious ones
HACE, high altitude cerebral oedema, is swelling on the brain. The signs are confusion, clumsiness, not being able to walk a straight line, severe headache that painkillers don't touch, and behaving oddly. The person affected is usually the last to know, which is why your guide watches how you walk.
HAPE, high altitude pulmonary oedema, is fluid in the lungs. Breathlessness at rest, a persistent cough, chest tightness, crackling breathing, blue-tinged lips, and exhaustion far beyond the day.
Both are emergencies. Both are rare. Both mean going down immediately, at night if necessary.
Descent is the treatment
Oxygen helps. Medication helps. Going down fixes it. Losing 500 to 1,000 metres usually produces a dramatic improvement, often within hours. Everything else your crew carries is there to make the descent possible, not to replace it.
What your guide does
Checks you twice a day, usually with a pulse oximeter on your finger, and asks about headache, appetite and sleep. The oximeter reading matters less than the pattern and much less than how you look and walk. A guide who says you're going down has the final word on that, and arguing with it is the worst decision available to you.
Your job
Tell them everything, including the symptoms you think are too small to mention and especially the ones you're worried will end your climb. A headache reported at lunchtime is a rest and some water. The same headache hidden until midnight is an evacuation.
What reduces the risk
A longer route, a slow pace, three to four litres of water a day, eating properly, no alcohol, and an acclimatisation walk in the afternoon. There's more on the mechanics in how acclimatisation works, and on medication in health and medication.