I'm to blame. The literal translation from the Latin is 'through my own fault'. Even those who don't speak Latin could probably make a guess that this phrase means 'I am culpable', or words to that effect.
The phrase originates in the Confiteor which is a part of the Catholic Mass where sinners acknowledge their failings before God. Confiteor translates as 'I confess'.
It has a long history of use in English and was used by Chaucer in his Troylus as early as 1374:
"Now, mea culpa, lord! I me repente."
To emphasize the point the phrase is sometimes strengthened to 'mea maxima culpa' - literally 'my most grievous fault'. This also has longstanding use, as here in Watson's Decacordon, 1604:
"Shall lay their hands a little heavier on their hearts with Mea maxima culpa."
The Confiteor uses both 'mea culpa' and 'mea maxima culpa'. In Latin:
Confiteor Deo omnipotenti,
beatæ Mariæ semper Virgini,
beato Michæli Archangelo,
beato Ioanni Baptistæ,
sanctis Apostolis Petro et Paulo,
omnibus Sanctis, et vobis, fratres (et tibi pater),
nimis cogitatione, verbo et opere:
mea maxima culpa.
Ideo precor beatam Mariam
beatum Michælem Archangelum,
beatum Ioannem Baptistam,
sanctos Apostolos Petrum et Paulum,
omnes Sanctos, et vos, fratres (et te, pater),
orare pro me ad Dominum Deum nostrum.
... and in English:
I confess to Almighty God,
to blessed Mary ever Virgin,
to blessed Michael, the Archangel,
to blessed John the Baptist,
to the holy Apostles Peter and Paul,
to all the Saints and to you, brothers (and to you Father),
that I have sinned exceedingly,
in thought, word and deed:
through my fault,
through my fault,
through my most grievous fault.
Therefore I beseech the blessed Mary,
blessed Michael the Archangel,
blessed John the Baptist,
the holy Apostles Peter and Paul,
all the Saints, and you, Father,
to pray to the Lord our God for me.
In everyday speech more recently mea culpa is used more lightheartedly and with little regard to its religious origin. The contemporary meaning when acknowledging responsibility is more like 'whoops, my fault'. Someone who admits 'mea culpa' may well be doing so in order to deflect criticism, as it could be construed as bad form to point out someone's failings after they have already acknowledged them, especially when the mistake wasn't overly important.
In recent years the US slang term, 'my bad' has been used to mean much the same thing.
See also - Latin Phrases in English.