One kilometre per hour is 0.6213711922 miles per hour. The factor is identical to the one between kilometres and miles, because dividing both sides by the same hour changes nothing — a speed conversion is a distance conversion in disguise.
About the units
Both units are a distance divided by an hour, and the whole difference lies in the distance. A statute mile is exactly 1,609.344 metres, fixed by the 1959 international agreement on the yard. The kilometre is a thousand metres. Kilometres per hour is the road speed unit almost everywhere; miles per hour survives in the United States, the United Kingdom and a handful of other places. Neither is an SI unit — the SI unit of speed is metres per second — but both are accepted, and no country posts road signs in metres per second.
The exact factor
The exact relationship is 1 mph = 1.609344 km/h, so 1 km/h = 0.62137119223733 mph. The speed limits that matter are worth knowing outright: 30 mph is 48 km/h, 50 mph is 80 km/h, 60 mph is 97 km/h, and 70 mph is 113 km/h. Going the other way, 50 km/h is 31 mph, 90 km/h is 56 mph, and 120 km/h is 75 mph. The Fibonacci trick from the distance conversion works here too — 8 km/h is about 5 mph, 13 about 8, 21 about 13.
Where you meet this conversion
Driving abroad, above all. A rental car crossing from Ireland into Northern Ireland moves from kilometres to miles at the border with no change in the road. Vehicle speedometers show both scales for exactly this reason, with the secondary scale in smaller figures. Weather reports give wind speed in one or the other depending on the country, and sports coverage of cycling, running and motorsport switches units between broadcasters covering the same event.