Socially competent people get along with others without difficulty; they know how to conduct themselves in any social situation, never feel themselves to be inferior to others,
and are easily able to put themselves in the other's shoes.
To be socially competent means to have social skills, to be good with people, usually in facet-to-face social situations. In normal use it would be opposite of being awkward, difficult, shy, disagreeable, and generally unable to function in the company of fellow human beings. For children, it would also normally include having knowledge and applying certain conventions, eg please and thank you, knowing what's appropriate in what situation, and towards what type of person.
You need to be careful with the word social. In fact, it means exactly what is implied by the previous posts, but there is another meaning coming into usage (probably from other languages, which use the same word): social more in the sense of societal. Thus you might hear of "social housing," which refers to low-rent or no-rent housing provided by the government (ie., the society) for people with insufficient income. There are other similar uses.
Socially competent almost certainly does not refer to social in this "socialist" sense.