In the base, acquired BEXP can be used in the place of regular combat EXP to help along the growth of a character. In Path of Radiance, units that receive BEXP level up like normal. However, in Radiant Dawn, units receive exactly three stat increases every time they level up via BEXP.
Bonus experience is acquired after clearing chapters and fulfilling certain conditions, the most common condition is completing a map under a turn limit which going beyond said turn limit will reduce the BEXP gained.
Tips for efficient use
Fire Emblem: Path of Radiance
You can first use Bonus EXP once you reach Chapter 8. You can use a little on Ike and Oscar, Kieran, or even Boyd at that time, but your best bet is to save it for two chapters and immediately pour as much as possible into Marcia. As a flier, she can make good use of it to instantly become a stronger, more mobile version of Titania.
Other popular targets include Jill and Astrid, for similar reasons, and Mist and Rhys, who can use it to quickly gain access to offensive magic (and a mount, in the case of Mist).
Fire Emblem: Radiant Dawn
Most players wait until their unit has maxed certain stats they normally would max out. For example, Mia normally maxes out Speed, Skill, and Strength. If the player has leveled her using BEXP, she would grow in other stats that she normally would not grow in. If the player waits to apply BEXP until a unit has at least one stat maxed, that stat will no longer raise when a new level is reached, therefore deferring to a less typical stat upgrade.
Additionally, while in the base the player can use BEXP to push the individual EXP of your preferred units to 99. By doing this, the very next action taken by the units will level them up. Since the level is in the context of a battle and is therefore considered an "organic" level, the level will not suffer from the three stat upgrade restrictions that comes with growing levels using BEXP.