Why is the recombination frequency higher if the genes are farther apart?
The genetic phenomenon referred to as recombination reflects the process of crossing over which occurs during meiosis. Crossing over creates an exchange of genetic information between homologous chromosomes. To a first approximation cossing over events take place at random positions along the aligned chromosomes. Consequently the further two loci are apart, the more likely that there will be a crossing over event between them. Thus the recombination frequency can be used to measure the distance between two genetic loci (or genes).
Supplement added in response to comment from OP
Imagine a piece of string. It has 6 marks on it (1-6 on the diagram) which divide it into 7 intervals (A-G). We are going to cut the string at one of the marks. We throw a die to decide which position to cut.
What is the probability that our cut will separate A and G? It's 100% since all single cuts will do this.
What is the probability that our cut will separate A and D? It's 50% since single cuts at 1,2 or 3 will do this.
What is the probability that our cut will separate A and B? It's 16.7% (1/6) since only a cut at position 1 will do this and the probability of throwing a 1 on our die is 1/6.
Think of the intervals (A-G) as genes and the cut sites (1-6) as possible crossing over events. If two homologous chromosomes are going to undergo a single crossover somewhere at random, then the closer together are the genes (intervals), the less likely is it that the crossover (cut) will take place between them. This assumes of course that the site of crossing over is chosen at random.
In the model presented here, if the order of the letters was scrambled and didn't know what it was but we were told the frequencies of occurrence of the separation of the intervals with random cutting we could deduce the order of the letters.