If I am not mistaken, I think it's protocol. The pilot must tell passengers what happened, or is happening, if anything out of the ordinary takes place mid-flight. It's not about scaring/not scaring passengers. It is about 'equipping' passengers with information - tell them what went wrong, reassure them by explaining what is being done to alleviate the situation, and calm them down, to avoid scaring them! Not telling them anything and letting passengers speculate will scare them even more, not to mention the potential of some passengers escalating the incident by panicking physically - because they don't have the right information! This is even more dangerous.
Of course, aircrews are trained to be a little more tactful when they have to make such announcements. They probably wouldn't say "We just lost one engine, and have to fly with just the one that's still working" but instead something more 'encouraging' like "Due to a malfunction, we have to proceed with the flight with number two engine - that's the one on the right-hand side, ladies and gentlemen" and proceed to tell what is being/will be done to land the plane safely etc. etc.
While it is safe for an aircraft to fly with one engine, certain conditions must be met - lower airspeed, lower altitude etc. And the aircraft is not allowed to taxi to the terminal on its own power and must stop as soon as it clears the runway upon landing. Engines (and the APU) must be shut down (to avert a fire hazard, I guess?). The aircraft must sit on the tarmac and wait for the aircraft tug to come out and tow it to a safe location where passengers can disembark. Imagine, if all this was happening and the pilot doesn't tell the passengers anything - what do you think will happen in the cabin? In serious, potentially life-and-death situations, people share and exchange real information, not play hide-and-seek with real information. I'm not a pilot nor an aviation expert, just an amateur AV geek and 'planespotter', but I WAS on a flight that had to shut one engine due to a malfunction and the events above did take place.
Scary? I think I was more tense than scared. And it was bl**dy unpleasant, but I'll get to this in a bit. It was a Malaysia Airlines flight from KUL - BKK. The pilot did a good job talking to passengers throughout the flight after the initial announcement. The passengers were calm. The aircraft flew very slowly and at a low altitude but other than that, there was no drama. The unpleasantness came when we landed. The aircraft stopped as soon as it turned off the runway and everything was shut down. This was on a midday in April and, sitting in the open under the Songkran sun in Suvarnabhumi, the cabin heated up very quickly. I think it was a good part of 30 minutes before the aircraft tug came to get us. During disembarkation nobody pushed, rushed or panicked. It was quite 'normal' - to the point where you get the usual impatient muppets flipping their seatbelts away, lunging for their stuff from the overhead storage and standing in the aisle - even before the aircraft has come to a complete stop. The captain made a final announcement, then came out and together with the crew apologised for the unpleasantness. But instead of rushing to get off, the passengers (including the impatient muppets) clapped and even stopped to shake hands with the pilots and crew and exchanged pleasantries.
I think this was all a part of their training and they achieved the desired outcome. I think they did a good job, don't you think so?