How did an inverted cross appear on Frenchie's neck?

Question

At the End of the movie The Nun, I was wondering how the cross appeared on Frenchie's neck.

I would also like to know what it means and what it can be there for.

Answer 1

how the cross appeared on Frenchie's neck?

During Sister Irene's and Frenchie's confrontation with Valak, it tries to kill Frenchie. That's when Irene comes to save him and Valak throws her away. Somewhere between this Valak possessed him.

Here is what Ed says during the end,

Take Maurice Theriault. His friends called him Frenchie. He was a French-Canadian farmer. Nothing more than a third grade education. Yet, after he was possessed...

After that, this scene appears.

(Ed continues) ...spoke some of the best Latin I'd ever heard.

Lauren: And like that, an upside-down cross started to appear from within his body.

So, he got possessed during that confrontation and that's why an inverted cross appeared on his neck.

I would also like to know what it means and what it can be there for.

This simply means that Frenchie is in demonic possession as inverted cross symbolizes demon according to all movies of The Conjuring universe.

It also points that the Frenchie is the guy who is shown in the beginning of The Conjuring, thus connecting the events of this movie to The Nun.

From Collider,

A quick shot reveals an upside-down cross—basically Valak’s house sigil—embedded in Maurice’s neck. Flash-forward 20 years to…a scene from The Conjuring. Ed (Patrick Wilson) and Lorraine Warren deliver a lecture on possession at Massachusetts Western University while Carolyn Perron (Lili Taylor) looks on from the audience.

Answer 2

According to Digital Spy:

It appears that Valak then strangles Frenchie to death before Sister Irene manages to seal the gateway by spitting the blood of Jesus Christ into Valak's face, sealing the gateway. However, Frenchie reappears and even though it seems that there's a happy ending, we see an upside-down cross on his neck and a flashback to Valak seemingly possessing him.

Frenchie's name is actually Maurice Theriault (based on a real-life case), the person we saw being exorcised in The Conjuring as the Warrens were talking about the three stages of possession: Infestation, Oppression and Possession.

The Nun then cuts to that very footage from the first movie (albeit with Bloquet's Maurice placed in) as we see Ed Warren talking about Maurice, the "French-Canadian farmer" who had a "dark spirit" possess him.

According to CBR.com:

As Frenchie parts company with Burke and Irene, the farmer reveals his actual name is Maurice, which is a major clue to his role later in the franchise. While the priest and nun leave Romania for good, Maurice scratches an itch on his neck, revealing to the audience the wounds he suffered while battling Valak are actually an inverted cross marking his own secret possession by the demon.

The film then flashes forward to a familiar scene from the series' first film, 2013's The Conjuring. Using footage from the original film, set in 1971, the scene revisits Lorraine and Ed Warren (Patrick Wilson) as they give a lecture on the nature of demonic possession, with Carolyn Perron, victim of the first film's demon, quietly in attendance. The Warrens describe a previous supernatural encounter with a farmer named Maurice who was possessed by a demon. The demonologists then show the class footage of them performing an exorcism on Maurice, with newly filmed material of Bloquet as an older version of Frenchie. Before being exorcised, the farmer touches Lorraine and lets out an evil howl, presumably beginning her connection to Valak years before the events of The Conjuring 2.