Learn Hardcore Russian: Мне удобно работать за компьютером у окна. - It is convenient for me to work at the computer by the window.

Russian often expresses feelings/states like удобно, холодно, жарко, скучно with a dative experiencer: literally To me it is convenient…
So мне удобно is the normal pattern. Using я would require a different construction (e.g., я чувствую себя удобно/комфортно, which is less common and sounds different).

What part of speech is удобно here, and why does it end in -о?

Here удобно is a predicative adverb/state word (often taught as a “category of state” word), used to describe a general state: it’s convenient.
The -о ending is typical for these predicate words (compare интересно, трудно, можно, нельзя).

Is there an implied это in the sentence?

Yes, conceptually it’s like Мне (это) удобно…, but Russian usually omits это in this kind of structure. Adding это is possible but often sounds heavier and more emphatic.

Why is работать an infinitive? Could it be a finite verb?

After predicative words like удобно, Russian commonly uses an to name the action that is convenient: , , .A finite verb would force a different structure, e.g. (a clause with ).