For centuries the horse had been a major form of transport throughout the world. This didn't end automatically with the coming of the motor car; indeed the German armies which invaded Russia during 1942 still relied mainly on horse power for transport. Once Europeans had actually invented the motor car, though, progress was rapid. The first dozen or so cars made in America were produced in 1896; within four years this figure had increased to around 8000, and 10 years later to about half 1 million. The horseless carriage was well on its way to world domination.
This is despite the fact that they were generally unreliable, expensive, and difficult to maintain. We take it for granted these days that if our car breaks down we can get new parts from a local stockist, and there is always a skilled mechanic available to fit them. At the beginning of the 20th century there was a huge variety of manufacturers, each with their own designs, and each one mainly producing it's own parts. Many of these manufacturers were short lived and so repairing broken car parts was a major problem for the owners. However, against that, they were cleaner (horse manure is not the most pleasant product), needed less upkeep and didn't need feeding and watering as often as the equine alternative!
Initially of course they really were 'horseless carriages' in that they resemble carriages, but were propelled by petrol, steam or even electricity from a bank of batteries. This was the way that things had always been, and there was already in existence businesses with the skills and materials to create these carriages.
They may have been viewed by many as easier to maintain than horses, but there were still huge problems for the owners to face on a regular basis. Breakdowns were so common as to be generally expected. Every owner really needed to become familiar with the car and it's expected faults, and be ready after a few seconds' notice to leave out the driving seat, get on or under the engine and try to fix the latest problem. Not least of all tyre wear and punctures were a constant issue since early tyres were nowhere near as resilient as they became in later years.
There was early opposition to the car; this mainly came from horse breeders were put out of business very quickly. Blacksmiths and carriage manufacturers, however, were often able to adapt quite quickly to the new fashion, and even stables could be converted into garage accommodation in which the vehicles could be protected from the elements, maintained and repair as necessary.
When Henry Ford came along and produced his Model T, the first mass produced car created on an assembly line, the motor car had truly come of age, and the time of the horse declined; but never disappeared completely.