Besmitten is simply an alternate version of 'smitten'; to be smitten (or be besmitten) with something or someone is to be enamored of them; to be in love, or a close approximation of it. It is somewhat cutesy and has archaic overtones, sounding like something that should appear in a Gothic novel from the late 1880s.
But it doesn't. Besmitten is a recent invention, and does not yet appear in any formal dictionaries, and very very few informal ones. (It does not, as of the time of this writing, appear on any of the major on-line dictionaries, including Urbandictionary and Wiktionary/Wikipedia). It was apparently invented out of whole cloth, copying the form of words such as beguiled, bedazzled, and besmirched. These days the primary function of the prefix 'be-' is as an intensifier, indicating that being besmitten is a bigger deal than simply being smitten.
If you are looking for a more formal word, 'besotted' shares the same meaning and actually appears in published dictionaries. However, besmitten is a commonly recognized word, and is perfectly cromulent in informal situations.
I have not been able to find who might have been the first to use the word besmitten, and it is likely at this point that we will never know. In Old English besmitan was the subjunctive plural form of besmite ('to smite'), although this has been out of use for hundreds of years. It is worth noting that the word besmit, while archaic, is still around in many dictionaries. It means to stain or contaminate, and so besmitten could obviously be used, if one wished, to mean stained or contaminated. But no one wants to use it that way.