What happens when star spins too fast?

The structure (mass versus radius and density profile) is influenced by its rotation rate, but not by as much as you might think.

Even in Newtonian physics you can think of a mass element $m$ at the surface of a star of mass $M$ and radius $R$, rotating with angular velocity $omega$.

A condition for stability would be that the surface gravity is strong enough to provide the centripetal acceleration of the test mass.
$$ frac{GMm}{R^2} > m R omega^2$$
If this is not satisfied then the object might break up (it is more complicated than this because the object will not stay spherical and the radius at the equator will increase etc., but these are small numerical factors).

Thus
$$ omega < left(frac{GM}{R^3}right)^{1/2}$$
or in terms of rotation period $P = 2pi/omega$ and so
$$ P > 2pi left(frac{GM}{R^3}right)^{-1/2},$$
is the condition for stability.

For a typical $1.4M_{odot}$ neutron star with radius 10 km, then $P>0.46$ milli-seconds.

Happily, this is easily satisfied for all observed neutron stars - they can spin extremely fast because of their enormous surface gravities and all are well below the instability limit. I believe the fastest known rotating pulsar has a period of 1.4 milli-seconds.

You also ask how pusars can attain these speeds. There are two classes of explanation for the two classes of pulsars.

Most pulsars are thought (at least initially) to be the product of a core-collapse supernova. The core collapses from something a little smaller than the radius of the Earth, to about 10km radius in a fraction of a second. Conservation of angular momentum demands that the rotation rate increases as the inverse of the radius squared. i.e. The spin rate increases by factors of a million or so.

Pulsars spin down with age because they turn their rotational kinetic energy into magnetic dipole radiation. However, the fastest rotating pulsars - the "milli-second pulsars" are "born again", by accreting material from a binary companion. The accreted material has angular momentum and the accretion of this angular momentum is able to spin the neutron star up to very high rates because it has a relatively (for a stellar-mass object) small moment of inertia.