Is "Fīliolō me" the ablative of the phrase or "me" refers to "me auctum" in the accusative? If is in the ablative, how does it translates?
This might be clearer if we add the implicit esse:
Fīliolō mē auctum esse scitō
(It's common to leave out forms of esse for brevity; they're seldom actually necessary.)
Now it's clear this is an indirect accusative+infinitive construction: the main verb is scitō, the accusative is mē, and the infinitive is auctum esse.
(literally) Know me to have been increased
(that is) Know that I have been increased
Since auctum esse is passive, it can take an ablative noun indicating who's doing the increasing: fīliolō, by a baby son.
Here are some possible translations into English:
Know that I've been enriched by a baby son
Know that my life has been made better by my new baby son
Me is accusative subject of the infinitive (auctum = auctum esse) in indirect statement triggered by the imperative form scito. Filiolo is ablative of means: 'Know that I have been enriched/enhanced by (provided with) a little son.'
For the record, this is from the opening of Cicero, Ad Atticum I.2.
Every student of Latin soon learns about the Ablative Absolute..
This is used when the Noun and the Participle have no position in the clause as Subject or Object. So in this Sentence the opening words are in the Ablative because they are "loosed from" (Absolute) from the clause, (confer: Nominativus Pendens)
L. Iulio Caesare, C. Marcio Figulo consulibus
However, if the Noun Participle phrase is the Subject it ought to be Nominative:
And, if it is the object of the verb (here= scito), then it will be accusative (here= me) and the participle will also be accusative in agreement (here= auctum).
Scito know me me auctum having been increased fīliolō by one little son.
--apologies for the mangled English.