It's all supply and demand really. If the demand is reduced, prices will fall in order for the excessive stock to be reduced.
In the long term though prices can not fall lower than the cost of raising the cattle. Farms with higher cost will be priced out of the market and will stop raising cattle.
This is all theoretical though. What is happening in the real world?
If we look at the statistics we'll see that beef retail price per lbs has almost doubled between 2002 and 2015 while production has fallen down around 15% and so have imports. In the meantime Americans are eating more meat
Around 7 million Americans are vegetarian, while that's significant, it does not see to affect the meat market that much.
So I think the conclusion is that vegetarians are not a driving factor of meat consumption and thus have minimal effect on prices and production either up or down. The number of vegetarians is probably somewhat stable so the chance of swings like the one you are asking about are minimal.
1. If you become vegetarian tomorrow that won't result in less animals being slaughtered. Your personal demand is just a rounding error in a huge industry.
2. If a million people become vegetarians tomorrow that might reduce the rate of increase of animals slaughtered, but the absolute numbers will not go down, the rest of the country will still outweigh the new vegetarians.
3. What if tens of millions become vegetarians? If it happens suddenly, the beef market will crash, the prices will hit rock bottom and a lot of farms will probably go out of business if the lower demand persists. Obviously the reverse could also happen, which is the bases of the claim being made.
Is any of this going to "save" any animals? If the demand is low, less animals will be bred for meat. Not sure if that qualifies.