I believe the full reasoning is that (1) if property owners paid property taxes on the full market value of their property, then (2) they would lobby for (or at least cease opposition against) upzoning for density, and (3) more dense housing would be legalized and built.
In most parts of the country, there's both an upside and downside to increasing property values to landowners. Upside: they're worth more, they could sell and move somewhere else cheaper, pocketing the money, or they could rent out their place for more. Downside: they have to pay higher ongoing property taxes. Most landowners still like increasing property values, but it's not all good for them.
With prop 13, the downside is wiped out. Increasing values are only good; they could increase 10x overnight and this would only make landowners filthy rich, there's no negative effect for them. Ergo, they have every incentive to keep housing supply constrained, to keep voting for politicians and policies that limit housing density.
And to bear specifically on the sub-point that blights the Bay Area -- by removing increasing property taxes, you are removing market pressure for highest and best use.
Another way to think about property taxes is that they're annual nudges to encourage best use of the property.
Without that pressure, as TulliusCicero notes, why would I ever sell before retirement? And furthermore, why would I even try to make money off the property (say, by renting)?
That's probably not in the city's (as in, all the people living there) best interest to have a substantial portion of homeowners so disengaged from the actual market.
PS: Which isn't to say there aren't gentrification and affordable housing debates to be had. Simply that Prop 13 doesn't really help with that -- it just removed risk from those who are already homeowners.
