There are two key concepts to understanding this sentence: the て form, and なる.
Function of -て
Firstly, the -て form of the verb is used to string events together, usually in sequential order:
I ate breakfast, and then took a shower.
When the second phrase expresses a passage of time, it describes how long it has (or had) been since the first phrase happened:
It's been a month since school started.
Thus, we can understand the first part of your example as:
(I) started learning English... (and then something else happened).
To a native English speaker, this kind of construction for telling time looks strange (yet logical), but it is very common in Japanese.
As for なる, it still is used with a noun here. なる is only used with adjectives and nouns. In fact, to use it with a verb, you have to turn the verb into a noun with よう or こと so that you can attach に. For example:
I became able to play the piano well
So in the second part of your (full) example, なる is used with the noun 半年, to indicate the passage of time:
It becomes a half year/half a year passes
The original sentence
So, putting these concepts together, we can decipher your sentence:
I started studying English, and then it became half a year. (lit)
I've been studying English for half a year.