'He sends her thousands of kisses' implies there are multiples of a thousand kisses (e.g. two thousand, three thousand, etc). 'He sends her a thousand kisses' states explicitly that there are one thousand kisses. In other words, 'thousands of kisses' means more than a thousand kisses, 'a thousand kisses' means exactly one thousand kisses. Hope this makes some sense.
I'm probably being stupid here, but why is it "lui le manda" and not "lui la manda"?
"She" is the indirect object of the sentence, not the direct object. The indirect pronoun is "le", while the direct pronoun is "la". It's the same as the difference between using "lo" (direct object) and "gli" (indirect object) for a male.
He sends a thousand kisses to her. --not-- He sends her (somewhere).
The direct object (the thing being sent) is the thousand kisses, not her.
Is this sentence right in English: 'He sends her thousands of kisses'? Can we say that rather than 'He sends her a thousand kisses.'?