Sister chromatids do NOT cross over. Non-sister chromatids from homologous chromosomes exchange genetic information (one chromosome from dad, and one from mom, for instance). This allows for the recombination of genes and accounts for one form of genetic variability in gametes.
chessxwizard got the concept, another way to look at this is to understand that the homologous chromosomes have different variations of the same alleles so when they cross over, shuffling occurs between dominant and reccesive alleles which has exponential variations in the combination of alleles a gamete will recieve after meiosis 2
the deal with crossing over is that it has the most effect on linked genes... so if the alleles for blue eye color and blonde hair for example are next to each other on the same chromosome, when they cross over there is a high chance that they will cross together and "replace" the alleles for brown hair, brown eyes on the other homologous chromosome so now the original chromosomes are all mixed up with different alleles from all over the place.
you want to ask yourself what would result if crossing over did not occur... that means the information between 2 homologs would NOT change and after 2 divisions of meiosis you would always result in 2 germ cells with the same info as the original homolog chromosome from which they came, and 2 other germ cells with the same info as the homolog chromosome from which they came.
if no crossing over occured, selective mating would still drive evolution because the combination of 2 genomes in the zygote would give varience, but crossing over just gives more varience.... hopefully this will give you another angle to understand why it happens.
they're not the "same"...they are very similar. this is prob the underlying reason you don't quite get crossing over. refer to above post (about eye color) for an example. if you still don't get it, feel free to PM me as i teach intro bio.
Homologues are formed by union of gametes: one homologue comes from mom, and another from dad. Each homologue duplicates during mitosis and meiosis, forming sister chromatids attached at the centromere. Synapsis occurs during meiosis and exchanges genetic information between non-sister chromatids of homologous chromosomes.