Two meanings:
1) "I'm up to my eyeballs in SOMETHING" = "I have to do A LOT of something".
eg, "I'm up to my eyeballs in work right now" = "I have SO MUCH work to do right now".
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Second Meaning:
Being "up to your eyeballs" in something can also mean that you are physically covered with something (usually liquid).
"I just fell into the lake. I'm up to my eyeballs in algae right now". = I just fell in the lake. I have algae all over me right now!
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ORIGIN:
It comes comes from being in water ... the water is so deep that it literally reaches your eyeballs. It is simply a metaphor in the sentences used above.
SIMILAR EXPRESSIONS:
"I AM DROWNING in work right now." = "I have SO MUCH work to do right now."
As usual, I hope this helps! Take care and good luck!