Mark Stevenson, lead artist of DK64, has already said in an interview with Nintendo Life that the use of the Expansion Pak to fix a bug was just a myth. This kind of bug did exist but was fully outside of the context of the Expanion Pak:

That story has become more-or-less accepted fact, although Stevenson believes the truth is more complicated. “This one’s a myth. The decision to use the Expansion Pak happened a long time before the game shipped, in fact we were called in by management and told that we were going to use the Expansion Pak and that we needed to do find ways to do stuff in the game that justified its use and made it a selling point. I think the bug story somehow got amalgamated into the Expansion Pak use and became urban myth.”

“There was a game-breaking bug right at the end of development that we were struggling with,” he clarifies, “but the Expansion Pak wasn’t introduced to deal with this and wasn’t the solution to the problem. My memory is that, like all consoles, the hardware is constantly revised over its lifetime to take advantage of ongoing improvements in technology and manufacture methods to essentially make the manufacture more cost effective and eventually profitable. I think there we’re something like 3 different revisions of the internal hardware by this point and the bug was unique to only one of these versions. We did eventually find it and fix it, but very late in the day.”

Source: Nintendo Life - Feature: Donkey Kong 64 Devs On Bugs, Boxing And 20 Years Of The DK Rap

Stevenson later in an Games Radar Interview explained that the Expansion Pak was used for having bigger maps as well as being able to do more advanced lightning techniques:

Artist Mark Stevenson remembers it being beneficial in terms of standard things like level size in Donkey Kong 64, but there were also more creative uses. “One thing I remember that we did use it for was that we had a lot of dynamic lighting in there, which was hard to do and expensive,” he recalls. “One of the engineers wrote a system whereby you’d go into a cave area, and there’d be a swinging light - the first swing of that light, it’d record all of the colour changes on all of the vertices in that area, and then save it as data and just play it back as an animation rather than going on to calculate the lighting constantly. You’d get a little bit of slowdown when you went in, but after that, it was nice and smooth.”

Source: Retro Gamer - How the N64 “confidently signposted our way into the 3D future”