Paedophile hunting seems to be on the rise (see e.g.: Paedophile hunters: Should police be working with vigilantes?, BBC News, 9 December 2016), and I'm surprised that it seems to lead to convictions (e.g.: Paedophile jailed after vigilantes set Bluewater trap, BBC News 21 October 2016). While what they're attempting might be abhorrent, it seems to me that technically all they've actually done is have a sexually explicit conversation with another consenting adult. What are they being convicted of?
Actual sex with a minor is an offence. Under the Criminal Attempts Act 1981, doing something "which is more than merely preparatory to the commission of the offence" is attempting to commit the offense, and is itself an offence. They have attempted the offence of having sex with a minor, although they didn't succeed because unbeknownst to them the other party was not a minor. There is some variability in whether it matters that there is no actual minor involved. In that specific instance the offence he was charged with was arranging the commission of a child sex offence (Sexual Offences Act 2003 §14), where one "intentionally arranges or facilitates something that he intends to do". Since he pleaded guilty, we won't find out how an appellate court would interpret 14(1)(b) ("doing it will involve the commission of an offence").