Where did the Goombas come from?

They came from the Mushroom Kingdom, being residents that joined Bowser.

A mushroom who betrayed the Mushroom Kingdom.
_{– Super Mario Bros. manual}

They are inspired by mushrooms. Specifically shiitake mushrooms:

We decided that that we should make an enemy that you could easily squash with a single blow. That's why we made it so that the first enemy the player would encounter would be a Goomba. But we created it right at the very end. [...] It's a shiitake mushroom!
_{– Iwata Asks: New Super Mario Bros. Wii}

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As an aside, in Japanese they are called Kuribō, which means something like "chestnut person". While presumably this refers to the colour, the design in Super Mario World was of a literal chestnut rather than a mushroom.

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_{click images to enlarge}