It is not just a new runway, but also an entirely new terminal and the infrastructure to support it, purchasing the necessary land around the expansion, putting a very busy road underground, and various environmental concerns.
This business jargon filled report from 2014 by Heathrow outlines the proposal, and after skipping past the normal buzzwords you can see their rough cost estimates on page 44:
£11.1 billion for:
Airport infrastructure
Includes runway, taxiway, apron,
aircraft stand, terminal capacity
and pier, tracked transit, baggage
system, access road, car park
and control tower costs
£0.9 billion for:
Surface access
Includes road, highway,
motorway, and rail costs
And £3.6 billion for:
Environmental
Includes reprovision of wildlife
habitat and flood mitigation
Community
Includes residential property,
commercial property, and general
land compulsory purchase,
community facilities re-provision,
community infrastructure levy,
and air noise compensation costs
Although the report is from May 2014, the total of these costs comes to £15.6 billion. That's roughly what the current budget for the expansion is, so this plan is likely close to how the money will be spent.