The UK was one of the largest members of the EU, and as such had a substantial say over it's direction and policies; and as the EU is a major economic power, consequently that is power in the world as well.

That is, it's government had this, as an "inside the room" participant. Power is often exercised by that means; in fact one definition of power is your having the means to sway others. For example, in the 90s the EU was debating whether to "deepen or widen", i e. to become more like a single state, or to admit more members. Germany and France led the faction that wanted to deepen, the UK, along with smaller allies, preferred to widen and admit more members from Eastern Europe. The UK's preference prevailed.

Because this mostly happened in diplomatic negotiations, it's not so well known, the press thought that EU bureurocracy was a better story.