Evolution occurs because biological organisms reproduce with variation, and variants reproduce at different rates.
Genetic variation is the result of genetic mechanisms such as reproductive recombination and spontaneous mutation. These mechanisms ensure that each offspring is a unique combination of alleles. Alleles are "rival" variants of genes with a specific "function". For many genes, having one allele rather than another will result in some difference in expression, resulting in turn in a difference in phenotype, which might result in a behavioural difference, a difference in appearance, or a difference in metabolic rate, for instance.
It stands to reason then, that it is possible that different variants react to circumstances differently. A variant with slightly longer intestines might find it easier to process certain foods, opening up a new niche for itself and its offspring. A variant with slightly longer fur might find it easier to venture into colder areas to gather food, again giving it and its offspring access to a new niche.
If the circumstances are thus that a particular trait or combination of traits makes it likely that a variant has a higher average number of offspring than its rival variants, then this will result in an increase of the number of alleles representing this variation in the next generation. This trend can, if these circumstances remain the same, persist until the great majority of the population possesses these successful traits.
Together, the emergence of new traits through reproductive variation, the spreading of successful traits and the decline of less successful traits through reproductive differential success, are called evolution, and the mechanisms behind these trends are the reason that evolution happens.