First of all, a CI for a given confidence percentage (e.g.95%) means, for all practical purposes (though technically it is not correct) that you are confident that the true value is in the interval.
If this is interval is "narrow" (note that this can only be regarded in a relative fashion, so, for comparison with what follows, say it is 1 unit wide), it means that there is not much room to play: whichever value you pick in that interval is going to be close to the true value (because the interval is narrow), and you are quite certain of that (95%).
Compare this to a relatively wide 95% CI (to match the example before, say it is 100 units wide): here, you are still 95% certain that the true value will be within this interval, yet that doesn't tell you very much, since there are relatively many values in the interval (about a factor 100 as opposed to 1 - and I ask, again, of purists to ignore the simplification).
Typically, you are going to need a bigger interval when you want to be 99% certain that the true value is in it, than when you only need to be 95% certain (note: this may not be true if the intervals are not nested), so indeed, the more confidence you need, the broader the interval you will need to pick.
On the other hand, you are more certain with the higher confidence interval. So, If I give you 2 intervals of the same width, and I say one is a 95% CI and the other is a 99% CI, I hope you will prefer the 99% one. In this sense, 99% CIs are more accurate: you have less doubt that you will have missed the truth.