I have some graph which has confidence interval. I'm trying to state something like "Error bars represent confidence intervals at the 95% level". Is this the correct way to phrase it? Is there a more standard way?
It you're asking for work that will be submitted to a particular journal or be reviewed by a particular group, you can always check what other works in that space do, as reporting CIs is a very standard thing to do. What you wrote is fine. For what it's worth, one conventional approach I see is to prefer "X% confidence interval" over "confidence interval at the X% level", so maybe "error bars depict/represent/denote the 95% confidence interval."
If your graph is a continuous plot (or close enough to continuous, so maybe a plot over time), it's sometimes common to refer to the CI as a confidence band, so in that case something common is to have the estimated trend, and then have the CI band be shaded some lighter color, and say something the effect of "the 95% confidence band is shaded in gray" or something..