Good question! It's indirect. For example, we can measure the circular orbit velocities of stars. Dark matter must have the same circular orbit velocity (i.e. velocity of a dark matter particle that is on a circular orbit), because it's subject to the same gravity. That's not yet the velocity dispersion, but it's close.

Next, we have observational evidence (e.g. rotation curves, lensing) that dark matter halos extend much farther than galaxies. This suggests that unlike the ordinary matter, the dark matter cannot efficiently cool -- otherwise it would condense into the galaxies as well. In fact the dark matter halos around large galaxies are consistent with what we find in simulations of dark matter that only interacts gravitationally. This suggests that dark matter is effectively collisionless in this context.

Since dark matter halos form by nearly isotropic collapse and accretion -- dark matter comes in from all directions -- their net angular momentum is small. Thus they should have very little net rotational motion and almost all random motion. This is also what the same simulations tell us.

The specific number "270 km/s" was a quick estimate I made by taking the isothermal sphere model, which is a good approximation for galactic halos over a pretty wide range of radii, and noting that its velocity dispersion is sqrt(3/2) its circular velocity. The local circular velocity is known to be 220 km/s, so that yields 270 km/s.