。
I'm so busy my head is spinning.
Literal
Eye [subject-が] spin [to-the-extent-of-ほど] busy is [emphatic-よ].
目が回る ('eyes spin') is the standard idiom for 'dizzy' — and by extension 'overwhelmingly busy.' The frame X ほど Y ('Y to the extent that X') uses the noun-like ほど as a degree marker — 'busy to the extent that my eyes spin.' The final よ is an emphatic/informative sentence-ender, flagging the statement as new information for the listener.