Let's say we are about a century in the future. The Earth's overcrowded, so natural resources fundamental to life as we know it (water, farming soil, energy, food) are not enough to meet everyone's needs (we can hypothesize we have evolved in a worldwide super-fair society, so social inequalities between different countries are non existent: we're all starving the same way).
The only way to avoid extinction is to transfer half of the world's population (let's say about 7-10 billion people, from a total of about 15-20 billion people) to another planet capable of hosting sustainable life.
It is a worldwide project: all governments, space agencies, research centers, major companies, share all of their resources in order to accomplish it and preserve humanity.
The other planets in and around the Solar System are inadequate: their gravity is either too low or too high for us to live, or can't be terraformed.
So we decide to engineer a planet like the Earth, with gravity, temperatures, atmosphere, soil chemistry and non-human life forms similar to those on Earth (and then bring all those people there).
We are in the 2100s, so the technology is quite better than the one we have in the 2010s, but nothing fundamentally different (e.g.: faster and larger spacecraft, larger and more efficient robotics, higher survival rates for serious diseases, but nothing like human teleportation or biological immortality).
Notice that, in order to transfer people efficiently, the planet must also be reasonably close to the actual Earth.
How do we do it?
If you like to think about it, more specifically:
What do we need to do it? Where do we start? Who's gonna do it (planning, design, manufacturing)? How long will it take?