In the Mazarin Stone, Billy the page is described thus: "Billy, the young but very wise and tactful page, who had helped a little to fill up the gap of loneliness and isolation which surrounded the saturnine figure of the great detective."

In HLV, in the Christmas kitchen scene, Bill Wiggins says "I’m his protégé, Mrs Holmes. When he dies, I get all his stuff, and his job." before amending that to "I help out a bit.", and Sherlock says "He’ll monitor their recovery. It’s more or less his day job."

So putting all that together with a little squinting...

Sherlock is recovering from his injury at 221B and feeling lonely and ill. He is now fully aware that he's in love with John but wants him to get back with Mary for the sake of the baby and because he knows John still loves her. So to stop John moving back in and tempting him, he invites Bill to live at 221B instead, ostensibly to teach him to develop the observational skills he displayed in Molly's lab and help him get off the streets. Bill basically becomes Sherlock's carer.

Then one day, Bill makes a clumsy pass at Sherlock, who thinks "well, why not?" and they start a casual sexual relationship.

Bonus points for the reactions of Sherlock's friends and family when they find out, given that Bill is poorly educated, not conventionally attractive and used to be involved in a drug den.