Just some comments to complement Joel's answer:

That forcing can destroy and then recreate measurability is due to Kunen:

Kenneth Kunen. Saturated ideals, The Journal of Symbolic Logic 43 (1) (1978), 65–76. MR0495118 (80a:03068)

The same is true for real valued measurability, this is due to Gitik:

Moti Gitik, Saharon Shelah. More on Real-valued measurable cardinals and forcing with ideals, Israel Journal of Mathematics 124 (2001), 221–242 ([GiSh 582]). MR1856516 (2002g:03110)

For a while it was open whether we can destroy and then reconstruct measurability while still preserving weak compactness. In all arguments I know it is essential that weak compactness fails in the intermediate model, as what we accomplish is a (destructible) instance of failure of stationary set reflection. Joel points out that his argument actually preserves weak compactness, and can be carried out in other settings to preserve measurability or stronger properties.

More generally, one can ask whether measurability can "appear spontaneously" by forcing, through some other means, just as we obtain saturation on the non-stationary ideal on $\omega_1$ by forcing MM, which cannot be traced back to some measure in some inner model that the forcing is reconstructing.

For another proof of the specific result you are asking (but one that destroys weak compactness in the intermediate extension), see section 4 of my paper on RVM cardinals:

Real-valued measurable cardinals and well-orderings of the reals. In Set theory. Centre de Recerca Matemàtica, Barcelona, 2003–2004, Joan Bagaria, and Stevo Todorcevic, eds.; Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, pp 83–120, MR2267147 (2007g:03064)

Just some comments to complement Joel's answer:

That forcing can destroy and then recreate measurability is due to Kunen:

Kenneth Kunen. Saturated ideals, The Journal of Symbolic Logic 43 (1) (1978), 65–76. MR0495118 (80a:03068)

The same is true for real valued measurability, this is due to Gitik:

Moti Gitik, Saharon Shelah. More on Real-valued measurable cardinals and forcing with ideals, Israel Journal of Mathematics 124 (2001), 221–242 ([GiSh 582]). MR1856516 (2002g:03110)

For a while it was open whether we can destroy and then reconstruct measurability while still preserving weak compactness. In all arguments I was aware of, it is essential that weak compactness fails in the intermediate model, as what we accomplish is a (destructible) instance of failure of stationary set reflection. Joel points out that his argument actually preserves weak compactness, and can be carried out in other settings to preserve measurability or stronger properties.

More generally, one can ask whether measurability can "appear spontaneously" by forcing, through some other means, just as we obtain saturation on the non-stationary ideal on $\omega_1$ by forcing MM, which cannot be traced back to some measure in some inner model that the forcing is reconstructing.

For another proof of the specific result you are asking (but one that destroys weak compactness in the intermediate extension), see section 4 of my paper on RVM cardinals:

Real-valued measurable cardinals and well-orderings of the reals. In Set theory. Centre de Recerca Matemàtica, Barcelona, 2003–2004, Joan Bagaria, and Stevo Todorcevic, eds.; Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, pp 83–120, MR2267147 (2007g:03064)

Just some comments to complement Joel's answer:

That forcing can destroy and then recreate measurability is due to Kunen:

Kenneth Kunen. Saturated ideals, The Journal of Symbolic Logic 43 (1) (1978), 65–76. MR0495118 (80a:03068)

The same is true for real valued measurability, this is due to Gitik:

Moti Gitik, Saharon Shelah. More on Real-valued measurable cardinals and forcing with ideals, Israel Journal of Mathematics 124 (2001), 221–242 ([GiSh 582]). MR1856516 (2002g:03110)

As far as I know, what is still the big open question here is whether we can destroy and then reconstruct measurability while still preserving weak compactness. In all arguments I know it is essential that weak compactness fails in the intermediate model. More generally, one can ask whether measurability can "appear spontaneously" by forcing, through some other means, just as we obtain saturation on the non-stationary ideal on $\omega_1$ by forcing MM, which cannot be traced back to some measure in some inner model that the forcing is reconstructing.

For another proof of the specific result you are asking, see section 4 of my paper on RVM cardinals:

Real-valued measurable cardinals and well-orderings of the reals. In Set theory. Centre de Recerca Matemàtica, Barcelona, 2003–2004, Joan Bagaria, and Stevo Todorcevic, eds.; Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, pp 83–120, MR2267147 (2007g:03064)

Just some comments to complement Joel's answer:

That forcing can destroy and then recreate measurability is due to Kunen:

Kenneth Kunen. Saturated ideals, The Journal of Symbolic Logic 43 (1) (1978), 65–76. MR0495118 (80a:03068)

The same is true for real valued measurability, this is due to Gitik:

Moti Gitik, Saharon Shelah. More on Real-valued measurable cardinals and forcing with ideals, Israel Journal of Mathematics 124 (2001), 221–242 ([GiSh 582]). MR1856516 (2002g:03110)

For a while it was open whether we can destroy and then reconstruct measurability while still preserving weak compactness. In all arguments I know it is essential that weak compactness fails in the intermediate model, as what we accomplish is a (destructible) instance of failure of stationary set reflection. Joel points out that his argument actually preserves weak compactness, and can be carried out in other settings to preserve measurability or stronger properties.

More generally, one can ask whether measurability can "appear spontaneously" by forcing, through some other means, just as we obtain saturation on the non-stationary ideal on $\omega_1$ by forcing MM, which cannot be traced back to some measure in some inner model that the forcing is reconstructing.

For another proof of the specific result you are asking (but one that destroys weak compactness in the intermediate extension), see section 4 of my paper on RVM cardinals:

Real-valued measurable cardinals and well-orderings of the reals. In Set theory. Centre de Recerca Matemàtica, Barcelona, 2003–2004, Joan Bagaria, and Stevo Todorcevic, eds.; Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, pp 83–120, MR2267147 (2007g:03064)