It's simple. You start with a satellite in geosynchronous orbit, so that it stays (more or less) fixed above one point on the Earth's surface. You hang a very strong cable down from it to that point, and a counterweight above it. on the end of another cable, because the centre of gravity (roughly speaking) of the whole structure has to stay at geosynchronous altitude. The all you need to get into space is an airtight vehicle that is attached to the cable and can pull itself up. The cost of the energy to do this is a tiny fraction of what rocket fuels now cost.
So what's the catch? Clarke mentions a couple of minor ones, namely orbital debris and stability against lateral oscillations; but the big one is the tensile strength of the cable.