Why does a flat universe imply an infinite universe?

We need to be precise about the phrase the size of the universe. Specifically I'm going to take it to mean the maximum possible separation between any two points. In an infinite universe two points can be separated by an arbitrarily large distance, so if the maximum distance between two points is finite this means the universe must not be infinite.

The point of all this is that the distance between any two points is calculated using the metric. For a Friedmann universe like ours (at least we believe our universe to be a Friedmann universe) the metric is (in polar coordinates):

$$ ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2d\Omega^2 \right] $$

The value of the parameter $k$ determines whether the universe is closed, flat or open. Specifically $k > 0$ is a closed universe, $k = 0$ is a flat universe and $k < 0$ is an open universe. The variable $s$ is the proper distance.

Now, suppose we choose an origin at some starting point, choose a fixed time, and calculate the proper distance, $s$ as we move radially away from the starting point. The question is whether $s$ can reach infinity or not. Because only $r$ is changing $dt = d\Omega = 0$, so the expression for the proper distance simplifies to:

$$ ds^2 = a^2(t) \frac{dr^2}{1 - kr^2} $$

We'll choose our units of distance to make $a = 1$, and we'll consider only closed or flat space, $k \ge 0$, in which case we can integrate the above equation to give:

$$ s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}} $$

So the maximum possible value for $s(r)$ is when $\sqrt{k}r = 1$, in which case:

$$ s_{max} = \frac{\pi}{2\sqrt{k}} $$

And there's the result we want. For a closed universe $k > 0$ and therefore the maximum possible distance between two points is finite. However as $k \rightarrow 0$ the maximum possible distance $s_{max} \rightarrow\infty$. That's why a flat universe is infinite.

However we should note that, As Rexcirus points out in his answer, even a flat universe can be finite if it has a non-trivial topology.

This claim is simply wrong. The flat hyperplane is of course infinite, but non trivial topologies can be flat and still finite. The simplest example is the 3-torus, but there are even the Klein bottle and the Hantzsche-Wendt manifold.

See for example page 27 of Janna Levin - Topology and the Cosmic Microwave Background, which show you ten different closed flat 3-manifolds.

For further reading I suggest: William Thurston, Three-Dimensional Geometry and Topology, Princeton University press (1997).

I think that it is important to note that (almost) everyone doing cosmology works within the framework of the FLRW universe.

This implies that we assume that the universe is spatially homogeneous and isotropic, i.e. 'every place is the same (at least on large scales)'. Now, think of a flat, finite universe: Is it possible to maintain that all places are the same?