Do people really steal invention ideas?

Unfortunately it happens all the time. Probably one of the most famous cases involved the invention of intermittent windshield wipers - luckily the inventor obtained a patent, and yet just about every major automotive company adopted the wipers on their cars without paying the invention a royalty or license fee. He sued and won but it took him literally decades to go through all of the appeals - I believe - I may be wrong - but I believe that eventually he died and his estate carried on the litigation - virtually every automotive company had to pay. So, one of the best ways to protect yourself is to file a patent application and seek a patent - you can initiate the process with a provisional patent application but it must be perfected within one year of the filing date of the provisional application by the filing of a non-provisional or PCT application. The main reason to obtain a patent is that it is an official document - it will obviously not necessarily prevent litigation, as I alluded to hereinbefore, but I believe that it is stronger than a simple NDA or other agreement.
Consult a patent professional.

It happens: in fact, I personally have suffered an eavesdropping industrial designer in a local, ostensibly-friendly incubator who took a concept (and even the working product name!) my partner was discussing, used it to raise money via Kickstarter, and acquired a patent for his (admittedly worse) version of the product. Of course, there is nothing legally problematic with what he did, and we went in another direction with our product, but the Busch-league industrial espionage experience was a stark reminder of the risks with working in communal shops. As already mentioned, it is most prudent to have at least a broad provisional application on file to lock in a priority date and, to the extent a receiving party is willing to execute one, a properly-drafted NDA that contractual prohibits the party from further disclosure or other use (including that which would not form the basis of an infringement claim) of disclosed materials and IP contained therein.

Yes, it does happen. An NDA doesn't necessarily stop it, but it allows you to take action if it happens. My main career now is as a consultant and expert witness in IP cases where I see inventions stolen. I've also had my own ideas and inventions stolen, but my trademarks and patents have allowed me to get compensation for the thefts.

The problem with most ideas (especially software) is the the value is market insight or figuring out or quantifying the needed. Actual R&D or development tends to be relatively simple compared to reaching and distributing the product/service. Because ideas are rarely unique, it is quite possible to leapfrog in execution ... and competition can come from anyone from customer to VC.