As a Resident Evil nerd (who has actually come to resent the old RE games), I can explain this. Publicly, Umbrella was a huge corporation who made everything from pharmaceutical products, cosmetics, and even computer technology and it can be assumed that they had a lot of dealings with various governments throughout the world. Unknown to the public, the company was founded by certain people who had other goals in mind. One of these people, Spencer had the goal of more or less creating an Aryan Master Race, which is how we got Wesker. Aside from from that, the deadly viruses they made were eventually sold to the highest bidders in the underground black market of.......deadly bioweapon trading. This included terrorist organizations and rouge governments.
This plot ended up the way it did because the first few Resident Evil games were inspired by b-horror/sci-fi films and didn't have stories which were meant to be seriously (a secret bio lab disguised at a mansion in the woods, come one). At some point, it was decided that the series should have a serious story and a continuity that works, so Capcom went from treating it like a b-horror film to treating it like a Mission Impossible film directed by Michael Bay.
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