Exit rate is a term used in web site traffic analysis and oil and gas production, as well as a financial term. Please, note there is a difference between exit and bounce rate.
Exit rate as a term used in web site traffic analysis (sometimes confused with bounce rate) is the percentage of visitors to a site who actively click away to a different site from a specific page, after possibly having visited any other pages on the site. The visitors just exited on that specific page.
Exit rate as an Upstream (petroleum industry) term refers to the rate of production of oil and/or gas as of a specified date. Often this will be the projected rate at the next year end.
Exit rate as a financial term refers to the revenue or cost to be expected in the following fiscal period as a derivative of the performance in the current period.
When used in the context of revenue, exit rate refers to the income expected in the following period as a result of sales closed in the existing period (assuming no further sales are made). If a company worked throughout the year and managed to signed up deals that will generate 1M USD a year in the following year (assuming no additional sales), then it would be said that company has 1M USD exit rate in this year.
When used in the context of costs, exit rate refers to the costs expected in the following period as a result of recurring costs taken on during the existing period (assuming no further additional recurring or one time costs are taken). If a company took on headcount and recurring costs of 1M/year during a given fiscal year, then the working budget for that company would have an exit rate of 1M USD in that year.