There were several reasons. One, everybody used two digit years for everything. When writing checks you used two digits. When you saw a date on television or in a book it used a two digit year. So it just made sense.

Secondly there was the idea that it would save space. And really it did. When your computer new out of the box had 3.5 KB available storage, those two extra characters were very important.

And thirdly because computers were new and things were changing so rapidly that everybody assumed everything would be replaced in just a few years. It's a bit difficult to explain, but electronic data was considered a lot "less real" then than it is now. When you switch the computer off it's gone. The idea that digital data would persist for years was difficult to grasp for a lot of people. Floppy disks had a realistic lifespan measured in months and later in years, but single digits for sure. It wasn't until optical drives became commonly available that it was possible for digital personal data to last very long at all.