I think that the key to understanding this is that the first $n-1$ rolls have already happened. This changes the event space that you need to look at.
A simple definition of probability is the fraction of “successful” outcomes out of the total number of possible outcomes. When you started off all of this die-rolling, all $6^n$ possible sequences of die rolls were in play. The event that consists of rolling a 1 every time is just one of these many possible outcomes, so its probability at that point is very small indeed.
After having rolled $n-1$ times, however, you’ve eliminated most of the possibilities with which you started. Now, there are only six possible outcomes instead of $6^n$, and so the probability of rolling another one is $1/6$. In fact, it doesn’t matter at all what the preceding $n-1$ rolls were: they’ve eliminated most of the possible outcomes that you started with, leaving only six.
Here’s an informal way to get a feel for this: When you start off, the chances of your rolling a one $n$ times are quite small, but with each successful roll the chance that you’ll actually do it improves. At step $k$, you only have to roll another $n-k+1$ ones, which is certainly more likely than rolling $n$ of them.