If you could visit the speaker as she's speaking this poem, we like to imagine that she's actually inside her own heart, rummaging around to find all the different kinds of love she has in there and counting them. Think of "How do I love thee? Let me count the ways" as a slightly more abstract version of something like "How many black shirts do I have? Let me count them all" – and then ransacking your closet, drawers, and laundry basket to get hold of each of them, noticing their differences and their similarities.
We get a sense of absent-mindedness here, too. How do I love thee? Wait, how many pairs of flip-flops do I have? Or, if you don't like the laundry analogy, think of the speaker's heart as her mp3 player, and she's spinning the little click wheel, reading off the titles of the songs she's got on there: "What kinds of music have I got on here? Let me count the songs. Oh, look, here's 'My Childhood's Faith' – I love that one." That sort of thing.