Evolution can be explained relatively simply: there are two basic mechanisms:

(1) Evolution: the change in hereditary traits in populations from generation to generation.

(2) Speciation: the reproductive isolation of daughter populations from the parent populations or other daughter populations.

Evolutionary Biologists sometimes call (1) microevolution and (2) macroevolution. Both of these are observed phenomena, and therefore facts on which we can base a theory:

Theory: these mechanisms are sufficient to explain the diversity of life as we know it, from the world around us, natural history, historical record, the fossil record and genetics.

This theory can be tested: compare it to the world around us, natural history, the historical record, the fossil record and genetics and see if there is some element of diversity that is not explained.

This theory can be falsified: all you need is evidence of some diversity that cannot be explained by these two mechanisms.

This theory can also make predictions:

From speciation we can predict that there are common ancestors, and the further you go back in time the more you will find common ancestors between existing life forms linking them into a tree of life, until you reach a point where original species lived.

From evolution we can predict that each lineage will show variation over time, that as you go back in time to those common ancestors that the lineages will become more and more similar until they merge with the ancestral parent population, and thus all evolution occurs within species.

That's it.

Enjoy.

ps - thanks Russ for the larger composing window.