While the answers so far give a good and correct answers, I'll offer my thoughts on the thinking process you may go through to gain more intuition. I would say the previous answers are obvious to most with a probability background, but I remember it took me several math courses before I could think of problems intuitively right away. Everyone starts on different levels and takes time to learn, but that doesn't mean you can't reach the intuition and appreciation for probability.
To gain intuition about why it wouldn't be close to 1, think about what happens when you roll a die many times. Your certainty that an ace shows up is never 1. Even if you roll the dice a trillion times it would still be 0.999999999... likely. While this may seem obvious, it's always important to remember that a finite number of rolls never hits 1 so you shouldn't just expect it to be close to 1 after only 6 rolls.
Another thing is the high chance that some faces will repeat themselves in the 6 rolls you do. If the faces didn't repeat themselves, only then would you have 100% certainty. Think about what happens when you take 6 balls out of a bag. If you put the ball back in the bag each time, there is a good chance you take out the same ball twice, and each time you do this, it limits your chance of getting the ball you are looking for. It just so happens that accounting for all of these replacement possibilities leads to the numbers you derived.
Generally when thinking about probability intuitively, you have to think about what are all the possible combinations. You shouldn't let large numbers intimidate your intuition because taking 6^6 is a big number. Factorials and exponential numbers are fundamental to probability, but we can use ratios to simplify the results instead of counting all possibilities in our heads.
A good approach is to think, what are all the ways in which I don't get what I am lookin for (ie, 1 ace). Think about the chance of getting multiple aces, multiple 1's and no ace, multiple 2's and no ace. When you add all of these up you get a much larger value than 0.01 (if you expected 99% probably). In fact, you get thousands, as mentioned in the other comments.