Like ことがある, ことができる is an expression that nominalizes what comes before it (turns it into a noun-phrase), and then highlights something about that (A) phrase. In the case of ことができる, the thing being highlighted about (A) is that it is 'possible'. This translates as 'can do (A)', or 'to be able to do (A)'.

• 魚（さかな）のように泳（およ）ぐことができるように毎日（まいにち）練習（れんしゅう）する。
  I practice everyday so that I will be able to swim like a fish.
• 彼（かれ）は料理（りょうり）をすることができる。
  He is able to cook.

However, こと is only required when the (A) phrase needs nominalization (is not a noun to begin with). Due to this, ができる may be used by itself when following a noun, without changing the meaning.

• 運転（うんてん）ができる彼氏（かれし）が欲（ほ）しい。
  I want a boyfriend that can drive.
• 力仕事（ちからしごと）ができる人（ひと）を探（さが）しています。
  We are looking for a person that is able to do physical labor.