This is a very introductory question to evolutionary biology. While the book "The selfish gene" is very pleasant to read and directly offer some key notions of modern evolutionary biology, it offers little to no introduction to the basics of evolutionary biology (it is not its goal). I highly recommend that you have a look at a short intro to evolution such as Understanding Evolution by UC Berkeley for example.
Below, I give a very brief answer to your question but taking an intro course to evolutionary biology would probably be more helpful.
What is evolution?
Evolution is a change in allele frequency in a population through time
An allele is a variant of a gene (or any other sequence at any locus). A locus is a position in the genome.
How do alleles come to existence?
Alleles are created from previous alleles, through mutations. The process of mutation is random in the sense that the effect of the new allele is "unknown from the organism" which create this allele. The mutation just occurs and it just has some effect on the phenotype and the fitness.
What are the forces of evolution?
There are a number of forces that affect evolutionary processes such as genetic drift, migration and natural selection for example.
What is natural selection?
Natural selection is the change of frequency of of the alleles at a given locus caused by differential fitness among the individuals carrying the different alleles.
Why one ought to expect that altruism should not exist in nature?
Here the term selfish and altruism are defined in terms of fitness effects. An altruistic individual is one that is performing a behaviour (or having other phenotypic trait) that is positively affecting the fitness of an individual of the same species while decreasing its own fitness.
Imagine an individual (let's call him the "actor") that has a mutation causing him to be altruistic (in opposition to being selfish). Such "actors" will have a low fitness while its neighbours will take advantage of the altruism of the "actor". There is therefore a differential in fitness between those that are selfish (higher fitness) and those that are altruistic (lower fitness) which ultimately yield to a decrease in the frequency of the alleles causing the altruistic behaviour until it completely disappears from the population.
The above paragraph explains why one would expect that altruism should be absent from nature. However, the story is definitely not that simple. But this extension is a story for another time. You will have to understand the different between "true (lifetime) altruism" and "false altruism", you will have to learn some game theory and finally Hamilton's extension of game theory.
What Dawkins mean by selfish gene
In his books, Dawkins talks about selfish gene often to refer only to genes that are causing meiotic drive (which might be confusing) and to talk about the existence of inter-genomic conflict. There are various mechanisms that can cause meiotic drive. You can learn more about that in Haid and Berstrom 1995, Jeanike 2001 or Jeffrey and Neuman 2002 for examples.