In origin, as we can deduce for its name, is a French concept. The idea was to amalgamate somehow the more powerful chest voice, and the weaker head voice. From here the 'mixte' (mixed).
One of the best examples we can find is Beniamino Gigli, in the beginning of Nemorino's cavatina "Quanto è bella, quanto è cara". Just listen to the smooth, light sound.
This technique has been related mostly to tenors, but also a baritone like Ivanov or a bass like Kipnis used something similar sometimes.
We can try to understand the concept better using a small fragment from Aida.
In the Third Act there is a beautiful duet between Radames and Aida, "Pur ti riveggo". Radames must reach a high A-flat in the phrase: il ciel de' nostri amori, on the 'i', that is quite difficult. Moreover, the score is marked as dolce. It must be a very delicate note, evocative, quite lyrical. How can a heroic tenor sings this note?. It just there, at the top of his tessitura. Will he use the 'voix mixte'?.
In the attached audio file, we can listen to 9 tenors.
Voix mixte - Aida
The first three give the note with their full voice: John Vickers, Franco Corelli and Giacomo Lauri-Volpi.
The second three tenors also attack with the full voice, but try to soften the sound, and look for the "mixte": Jussi Bjorling, Carlo Bergonzi and Gigli himself.
The last group go for the "mixte" from the beginning: José Carreras, Max Lorenz and Pavarotti.
Listening to this, we can figure out what "mixte" is, and also decide our favorite approach for the passage. Personally, I go for Max Lorenz.