The phrase is often used to mean that someone is eager to please a person. To be at someone’s ‘beck and call’ is to be ready to do something for someone at any given time.

The ‘call’ part is fairly obvious, if someone calls, you’ll answer. But the ‘beck’ part seems fairly strange, that’s because it’s a word we don’t really use anymore, it was used as shorthand for ‘beckon’ in the late 14th century but has now fallen out of favour.