〜のが好き・上手
like / be good at doing
Verb dictionary form + のが + 好き / 上手 / 下手
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好き, 上手 and 下手 need a noun before が. To use a verb there, turn it into a noun with の: 本を読むのが好きです, "I like reading"; 料理を作るのが上手です, "(he) is good at cooking".
Think of 読むの as "reading": the dictionary form plus の is a noun made from the action.
こと does the same job and you'll see it too (読むことが好き); の sounds more everyday, こと a little more formal or abstract. Both are fine.
The が stays, because 好き is still an adjective: 読むのを好き is wrong.
Examples
本を読むのが好きです。
I like reading.
兄は料理を作るのが上手です。
My older brother is good at cooking.
朝早く起きるのが苦手です。
I'm not good at getting up early.
泳ぐのが好きですか。
Do you like swimming?
私は絵を描くのが下手です。
I'm bad at drawing.