Notice Definitions 3,4,5,6 http://www.thefreedictionary.com/heart
All pointing to "heart" in a metaphoric sense as the seat of things like emotion, strength, and courage.
To "have a lot of heart" means to have a lot of these deep inner resources.
(and is not limited to compassion as some might assume)
This metaphoric use of "heart" has a long tradition in English.
Great Hearted from the 14th century, is almost identical to "a lot of heart."
Several common English words are borrowings from other languages that already used this same metaphor. The French word for "heart" is "cœur" whose root can be found in courage, encourage, and discourage.
A few other "inner workings" have similar metaphoric meaning, like Nerve & Guts