If dominant genes override recessive genes, how do recessive genes still exist?

That is incorrect. Any parent with a recessive allele will can expect on average to pass on that allele to 50% of their children, regardless of what alleles their partner has.

Now, that allele may not be expressed in any of their children depending on their partner's alleles, but that's different from transmission. A kid who has the allele but doesn't express it still has the allele and can also expect to pass it on to 50% of their own kids.

Maybe you are under the impression that only beneficial genes get transmitted ? In which case a recessive allele not being expressed (and therefore not given a chance to be beneficial) would be a problem. But that's not the case. Alleles don't even need to be beneficial to take over the population ! The baseline way an allele spreads or not if we assume its effect is neutral is a random walk: every generation has a bit more or a bit less of the allele than the previous, depending on the luck of the draw in terms of which of their two alleles parents passed down to their kids, and this leads the percentage of the population with the allele to go up and down randomly until it reaches 0% or 100%, after which it stays there (either because it's no longer there to be passed on, or there are no other alleles to be passed on instead of it). Mathematically it can be shown that given enough time, one allele must necessarily take over the population simply because a random walk going on for infinite time passes through every possible point, which means every allele will at some point hit 0% until one is left at 100%. The odds of any given allele eventually taking over the population is equal to its share of the population (so an allele that's in 10% of the population has a 1 in 10 chance of being the one to eventually take it over).

Of course in practice there are new alleles appearing all the time, and the maths vary when the population size changes and such. However the key thing to understand here is that whether an allele is beneficial or harmful doesn't really change the basic dynamics of how it spreads, it just nudges the probabilities - a beneficial allele is more likely to increase than decrease in frequency every generation, and a harmful allele does the opposite. This means that the odds of a beneficial allele taking over the population is bigger than for a neutral one, and the odds of a harmful allele taking over the population is virtually nil unless it's practically neutral.

For a beneficial gene to be recessive means in practice that it's more weakly beneficial than if it were dominant - there is still a benefit because there's a chance some of your kids and grandkids will be heterozygous for it and reap its benefits, but it's clearly less of a benefit than if it were guaranteed to express every time. However that still puts it at or above the neutral baseline, and neutral alleles get passed on just fine.

I think you might be getting hung up on the difference between the expression of a gene and its inheritance. Dominant alleles of a gene express their phenotype over recessive alleles of a gene but both alleles have an equal chance of being passed on to offspring.

There's a 25% chance that any single progeny from two parents who are heterozygous (Aa) will be homozygous recessive (aa) but a 75% chance that any single progeny will have at least one recessive allele (Aa or aa). This means even for rare recessive phenotypes the recessive allele may be fairly common in the gene pool.

The last thing to keep in mind is that the concept of alleles is slightly different than the concept of genes. Two people may both have recessive alleles but they may still have small differences in the exact sequence of their genes, so while they both express the same phenotype and have the same allele genotype their genes aren't exactly the same.