The meaning is “with sincere and deep feeling.” Two ideas on the origin, both intuitive rather than scholarly: (1) People often feel emotion in their chests, specifically in a place that’s about where the bottom of the heart is. (2) When thinking of the heart as the seat of emotion, people may conceive of it as a (metaphorical) container that fills up with feeling, so that “from the bottom” means the heart is filled through-and-through with love or gratitude or whatever, not just partly filled. (Think of the bottom of the inside of the heart, not the bottom of the outer surface.) The opposite kind of imagery appears in “off the top of my head,” meaning that an idea is superficial.

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