So large I am, you could not see,
Bigger than aught that you could dream.
Infinity is larger than anything you can imagine. This is self-explanatory.
I ate so much, I must lie down,
To lay the weight off you again.
The ∞ symbol is a sideways 8 ("ate").
"To lay the weight off you again" uses wordplay: "you again" is W, and removing the W from "weight" gives "to lay the eight". This is again referring to the symbol being a sideways 8.
You might think there's one of me,
A counter counted: there's more, it seems!
This refers to Cantor's proof of multiple sizes of infinity (in particular, the natural numbers and real numbers have different cardinalities, and so the real numbers are uncountable).
Lemon is Kate, by sight and sound,
But I can't fit inside your brain.
The ∞ symbol is called a "lemniscate", which sounds like "lemon is Kate". The last line is essentially the same as the second.