寒くて is 寒い in the 〜て form. It’s kinda hard to explain what this does because it’s so flexible but here goes…
I think the simplest way of explaining it is that it links 「今年の冬はとても寒い」 and 「（私は今年の冬を）大嫌い（だ）」 together. I’ve added the implicit bits of the second clause in brackets so that you can see the underlying structure.
In this case, since we have a statement about the weather and then an opinion, it implies that hating this winter is a linked to (i.e. is a consequence of) this winter being cold.
There’s too many other ways to use it to explain here, but it’s a really useful pattern so it’s worth learning.
Awesome explanation, thank you, it was not included in Tae Kim’s guide, or probably I missed it, but I doubt it… it means I still need to learn a lot about the grammar!!.. nice nice, 好き!
In addition to just implying the connection between the statement and the opinion, it puts more emphasis on the emotional reaction when you use the て form.
Something like: 「今年の冬はとても寒いから大嫌いよ」sounds far less natural than Vegpie’s て construction. 大嫌い is quite a strong emotional response, and so linking the two clauses together with て really brings out the emotion of the sentence. “It’s cold this winter, and I REALLY HATE it!”
For another example:
「試験に合格してよかった！」 - “I’m glad I passed the exam!” (sounds really happy)
「試験に合格したからよかった。」 - “I’m glad because I passed the exam.” (sounds like a robot)
Edit: Oops, I meant for this to reply to Ollylove/OP and not Vegpie - sorry about that!
