How do scientist know the size of the universe?
The scientists do not know the size of the Universe.
What scientists know is the size of the observable universe. This corresponds in theory to an object that emitted light at the time of the Big Bang. Anything farther than this, its light hasn't arrived yet, so it is impossible to be known.
So, the farthest things we can see were at the distance of about 13.7 billion light-years from us, by the time of big bang.
But since the universe is expanding those objects are now at about 46 billion light-years from us.