Why did a lot of people think that the world was going to end because of Y2K?

Nobody thought the world would end. The concern was that many electronic devices would quit working and this would have cascading effects. For example, if someone's electronic life support system hung because it had a two-digit year field and this caused it to hang, the person could die. Ditto for control systems on airplanes, trains and more. The worry was that electronics would stop working and we'd have no notice, and the effects could have been disastrous.

yes just make new computers for the most part

as for the world ending... people see that all the time ,, like when hostess went out of business and no more twinkies

But for some of use with Windows 10 my DOS programs will not run

it is not the end of the work but it represents the end of people understanding real bookkeeping and move to putting numbers in boxes and let the computer rule your life.....see SIRI and Elexa

It is the "spiritual" power of that big number. Just as in the run up to AD 1,000 much of Europe (which was more "Christian" in those days) awaited the Second Coming of Jesus.

There was also the thing about the "Millennium Bug" in which it was supposed that many computers would crash because the year dates in early computer programs had just 2 digits (95, 96, 97, etc) and thus nobody could predict what might happen when "00" or "20" came up - would those dates be seen as previous to 96, 97, etc?

It's not that "the world was going to end". But if all the computers had stopped working, that would have included things like the power grid, air traffic control, telephones, the stock market, records of all kinds, etc. It would have been chaotic.

because a lot of people are stupid work

People didn't think the world would literally end, but it could have caused chaos.

Y2K was a programming issue which was going to crop up with the start of the year 2000. At the time many computer programs, in order to save space, used only two digits for the year. So 1988 would be rendered as 88. This could cause a problem when the year rolled over to 2000, because computers might read that as the year 1900. That might cause any number of problems with some processes producing bizarre results and some programs crashing altogether as they couldn't figure out what to do. This wouldn't literally "end the world" but might cause headaches, possibly significant ones. Bank computer programs might cease to work, leaving patrons with little ability to withdraw money or use credit cards, paychecks might be late, monitoring systems might malfunction. The results could be serious, especially if these all hit at once.

The early computers had very limited memory, so to save memory usage the programmers dropped the first two digits of the year. So, instead of having the date format 1/1/1990, the format was 1/1/90. By the late 1990s computers were ubiquitous, running electrical grids, water supply systems, banking systems, air traffic management systems, just to name a few. Of of the functions of these computer systems was to keep track of the elapsed time.

The problem was that at the stroke of midnight on December 31, 1999 the calendar would reset to 1/1/00 and the timekeeping functions may have crashed. While the companies, over the years, upgraded their software, there was still plenty of legacy software running in the background and nobody knew how it would affect the computers running it.

To keep their operations running normally through the date roll over from 1999 to 2000 all the governments and businesses had to make a detailed analysis of their computer systems and the software or upgrade to the entirely new ones. This effort lasted for several years through the second half of 1990s.

No. The financial systems would have crashed. The power and water would have gone out--people would have DIED. If teams of coders didn't spend 6 months not sleeping fixing it, that would have happened. It enrages me the lesson people try to suggest was "y2k was fake/people are stupid" -- no, the lesson is we can fix problems if we bother to try. BTW, they only fixed it until 2038 when the same bug will reoccur. Nobody is working on fixing that. Why bother, they have time! Right?

A lot of people (civilians) thought that comuters in missile silos, on board ships and subs would go crazy and launch missiles and what not. In fact a computer can not and never could simply launch an ICBM on its own. It takes people to turn the launch keys.

Having a system that would allow a computer to launch missiles on its own would be incredibly stupid...

Simply: they believed the (uninformed) hype.