Lets start out with Hayek's motivation when writing the book. During the War Hayek lived in England and had a lot of contact with British Intellectuals. He himself was a very well known and respected political economists (we call it simply economist).
During this this, the British intellectuals were turning more and more socialist. This had been a important part of British intellectual live for some time. Fabian socialists believed that you could have a fair socialism that was not as disruptive or dictatorial like Bolshevik Socialism in Russia.
Because a War Economy is very much like the imagined Socialist Economy, many intellectuals just wanted to keep going in the style of the war economy. Hayek thought this was a extremely bad idea and he wrote 'The Road to Serfdom' to explained to them that it is not as simple as the thought it would be. He tried to show that planning could not go hand in hand with democracy and individual freedom. His goal was to moderate British intellectuals in how far they would go, in implementing their ideas.
The book completely discredited him professionally. It was one of those books that you know was evil, even if you had not read it. Many important intellectuals flat out refused to read it. However the book did have the intended effect, a number of important people read it, and spurred a lot of discussion. Many people who were more moderate, were reminded that their were certain lines that they would not cross. It also lead to everybody clearly stating that the most definitely did not want to end up in a Stalinist situation.
The fear of Statism taking hold was seen not only by Hayek, but also by Orwell in his 1984. Orwell attacked English Socialist who he thought were to willing to take it to far, by naming the evil party in his novel Ingsoc (English Socialist Party).
I have to remind you that during this time, Soviet were seen as good and strong by a lot of people. They argued that the Soviet system had worked brilliantly and it would be good for England to be like the Soviet, just with more democracy.
Now, as I have explained, "The Road to Serfdom" was a book for english intellectuals and to directly address them. Hayek never thought this book would sell well in England or anywhere else.
However when in 1945 a Reader's Digest version was released in the US it turned out to be a bestseller.
The ideas that made Hayek unpopular in England found more footing in the US. The book influenced many US economists and helped a movement of people that were against such socialist policies that were happening in England.
Hayek was a believer in the power of ideas. He always believed that the most important thing was to convince people that socialism and more government planning was not the way to solve all problems. If the goto answer for every problem was government intervention then you were on a slippery slope.
He wanted people to come up with their own ideas and solution, to solve problems between them selfs as far as possible. Hayek believed in a great many functions for the government (specially in 'The Road to Serfdom') but he wanted to convince people that government action was the last solution that should be considered.
Today we face very much the same arguments as back then. How much should the government make your choices. Should the government be allowed to regulate cup sizes in fast food restaurants? Should government be allowed to stop you from doing heroin? Should government provide health care for everybody? Should they only run insurance, or should the run hospitals as well?
The Road to Serfdom is on of the defining books in this argument that has shaped the last century like no other argument has. The argument is not over, Hayek's writings provide one side of this argument and this is why he is still read today and this is why he is still influential today.
He should be read, together with the other great social thinkers of the last century (and the centuries before that as well).