に気（き）が付（つ）く is a common expression that highlights when someone ‘notices that (A)’, or ‘realizes that (A)’. 付（つ）く ‘to be attached’ is the intransitive form of 付（つ）ける ‘to attach’, meaning that it is something that happens by itself. The 格助詞（かくじょし） (case-marking particle)
に marks the ‘location’ to where your 気（き） (sometimes interpreted as ‘attention’) becomes attached.
This structure may be used before nouns, or verbs that have been nominalized with either
の, or
こと.
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待（ま）ち合（あ）わせ場所（ばしょ）に着（つ）いてから、携帯（けいたい）を忘（わす）れたことに気（き）が付（つ）いた。
I realized that I had forgotten my phone only after I arrived at the meeting place.
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犬（いぬ）に餌（えさ）をあげていないのに気（き）が付（つ）いた。
I realized that I didn't feed my dog this morning.
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猛（もう）スピードでこっちに来（き）ている車（くるま）に気（き）が付（つ）いた。
I noticed that there was a car heading this direction at a crazy speed.
There are many different translations for 気（き） in English. These can range from ‘spirit’, to ‘attention’, to ‘will’, or even ‘consciousness’. However, perhaps one of the most accurate ways to translate 気（き）, is ‘presence of mind’. The ‘location’ to which your focus, attention, or perception is drawn, will always highlight your 気（き）.