Exit rate as a term used in web site traffic analysis (sometimes confused with bounce rate) is the percentage of visitors to a page on the website from which they exit the website to a different website. The visitors just exited from 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.
When used in the context of revenue, exit rate refers to the income expected in following periods as a result of sales closed in the existing period. If a company worked throughout the year and signed deals that will generate a million dollars a year in following years, then the company has a one million dollar per year exit rate in this year.
When used in the context of costs, exit rate refers to the costs expected in following periods as a result of recurring costs taken on during the existing period. If a company took on headcount and recurring costs of a million dollars per year during a given fiscal year, then the working budget for that company would have an cost exit rate of a million dollars per year in that year.