To increase a firm's amount of debt. In general, a firm leverages up by issuing a bond, often in order to finance an expansion of operations. A publicly-traded company may leverage up to repurchase its own stock, which usually increases the share price; this was a relatively common way to discourage hostile takeovers in the 1980s.
To increase the portion of debt in a firm's capital structure by issuing debt and using the proceeds to repurchase stock or by financing any new expansion through debt. In the mid-1980s, firms decided to leverage up in an attempt to improve the market price of their stock, thereby fending off takeover attempts.