できる in the sense of "to be able to do something" is a little unusual as verbs go, in that what you would normally think of as the object (the nominalised action that you can do) always takes the subject particle が. できる never takes the particle を, and is used as follows:
私は数学ができる。 I can do maths.
私は平泳ぎができる。 I can do the breaststroke.
私は高く飛ぶことができる。 I can jump high.
The reason behind this is probably because できる is originally an intransitive verb with the basic meaning of "to come into existence", "to be made", "to take shape". It's still commonly used in this sense, in sentences like:
新しい城ができた。 "A new castle has been built."
車は速く動くようにできている。 "Cars are made to move fast."
This sense can also be extended to less literal usages, such as:
私は彼氏ができた。 "I got myself a boyfriend."
(literally: "As for me, a boyfriend came into existence.")
And I believe the sense we're interested in here, of "being able to do something", is actually an extension of this kind of usage.
私は平泳ぎができる。 "I can do the breaststroke."
(literally: "As for me, the breaststroke takes shape.")
私は高く飛ぶことができる。 "I can jump high."
(literally: "As for me, the thing of jumping high takes shape.")
Of course, nobody really thinks of it in this way - for all practical purposes, できる is used in the same way as the English "can do". But because of these roots, grammatically it takes the action itself as a subject, not an object.
The relationship between Japanese できる and English "can do" in this usage is actually almost identical to that between Japanese ある and English "have". In both cases the root meaning of the verb is intransitive (できる = "to come into existence", ある = "to exist"), but they are also used to express a meaning that is expressed with a transitive verb in English ("to be able to do" and "to have"). In both cases the object of the relevant verb in English becomes the subject in Japanese:
彼はお金がある。 "He has money."
彼は数学ができる。 "He can do maths."
Whilst what would be the subject in English - the person who has the object / is able to do the action - can either be the topic of the sentence, or alternatively can be stated as an indirect object with に. In basic sentences this に will generally also be topicalised:
彼はお金がある = 彼にはお金がある
彼は数学ができる = 彼には数学ができる
But the に can also occur on its own, for instance in questions:
彼に何がある？ "What does he have?"
彼に何ができる？ "What can he do?"