Ok, let's do this:
If you wrote it in kanji, which you don't, it would be 御座います. 御 (ご) is the same honorific you find in ごはん, the chinese to お. 座 (ざ) means, among other things, position, status. So you get "oh honorable status".
Taking in consideration that ございます is used in polite speech when you put yourself beneath the person you're talking to, it kind of makes sense.
(Also there is a verb 御座る that is not really used anymore that means to be)
(Also#2 this was a quick research on jisho because I got curious too, if someone has a deep historical meaning to add, please do?)
ございます is the verb ござる with the honorific ます. In the Kyoto dialect, historically り became い in some contexts, including verbal inflections, which is why this isn't *ござります. Since, during the Edo era, Kyoto was the cultural center of Japan, many Kyoto-dialect features were borrowed into the Edo/Tokyo-based standard. おはよう is another good example. The ending -く regularly became -う in that dialect, and はやう became はよう (au → ō was a regular change).
Ahh, that first (Also) has answered a question for me. Lots of history nerd anime characters use the archaic "gozaru."