The article is about salary negotiation - the practice of influencing your own salary either when taking a job or during a pay review.
When speaking about salaries, especially high ones, we often quote the number of figures in the annual salary, for example:
£20,000 is a five-figure salary
£200,000 is a six-figure salary
Even though there is an immense difference between someone who earns £15,000 a year and someone who earns £80,000 (yet these are both "five figure" salaries), there is thought to be some status in someone who can boast a "six figure" salary.
Your question is about the phrase:
one of their consultants would chop five figures off their own invoice with an email.
Without any context it is clear to me it is speaking about saying something in an email which would cost you five figures - the different between receiving £2,500,000 (7 figures) and £25!
Of course, if you read the whole article for the context the answer is right there - the writer is referring to a specific example he quoted earlier:
I cost myself five figures with a single email. The particulars are boring, but suffice it to say I fairly recently made a wet-behind-the-ears-engineer error in quoting a client. He noticed. So did my bank statement.
So apparently the writer made an error billing a client in an email which resulted in the payment being five-figures less than intended.