When you attach your energy (imagine someone suddenly looking up at something, attaching their energy to it), you have noticed or realized something. That's why this word means to realize or to notice.
Don't confuse this with 気を付けて which is similar in that it has all the same kanji. It's just used a little different. Try to remember the difference before moving on.
The readings come from the 気 kanji (which you know) and the word 付く which you should know by now. Combine these together and you have the ability to read this word. Take note that the 付く part is rendaku'd, though, so it is a tiny bit different.
to notice me
to become aware of feelings
to notice that matter
Don't pretend like you didn't notice.
I think I realized my feelings for Koichiko three days after we met.
I can’t believe you gave me your ice cream. I didn’t realize that you were so kind until now.