When and how did the criminal sense of 'grooming' arise?

It almost certainly comes from slightly older senses of groom, meaning to tend and prepare.

In the words of the OED, "groom" can mean:

To tend or attend to carefully; to give a neat, tidy, or ‘smart’ appearance to.

and more relevantly:

To prepare as a political candidate; in extended use, to prepare or coach for a career, a sporting contest, etc. Originally U.S.

The former is traced to 1843, and their examples of the latter go back to 1887. The source of this idiom is obviously from grooming a horse, which means to tend, feed, brush, and look after an animal.

The OED has a draft addition under the entry for groom v.

Of a paedophile: to befriend or influence (a child), now esp. via the internet, in preparation for future sexual abuse.

This goes back to 1985:

Chicago Tribune 28 May v. 8/2 These ‘friendly molesters’ become acquainted with their targeted victim.., gaining their trust while secretly grooming the child as a sexual partner.

They don't give a specific description of how the meaning has shifted but it seems reasonable to suppose it has gone from meaning to tend and prepare a horse for use, to meaning to tend and prepare a human being for a purpose.

"groom, v.". OED Online. December 2020. Oxford University Press. (accessed February 17, 2021).

"Grooming" has multiple meanings. In the context of career, the meaning is metaphorical of improving appearance. In the context of sexual abuse, I believe it is drawn more from that of primate social interaction. Grooming among primates appears to have many functions, including as a prelude to coitus, but all of them relate to social cohesion. So as it relates to sexual abuse, it is any action that establishes or advances the (inappropriate) relationship.

To answer the final question:

Is it an AmE usage?

No, it's not just North American - it's well understood in Britain as well, without need for any adjective in context (as in your sample).

I have two Oxford dictionaries to hand: ODE (3rd ed, 2010) lists this meaning (as part of the sense to prepare or train someone), though my slightly older SOED (5th ed, 2002) does not.