look into: meaning and explanation
to look into sth = to investigate sth.
When you look into something (such as a crime, a problem or the possibility of something) you are trying to discover or find out more information about it.
For example, before making a big purchase (such as a car), you probably spend some time looking into the advantages and disadvantages of different models, or how much different cars cost.
Have a go at these micro-dictation exercises to hear this expression being used in context – how much can you understand?
About the sentence
…she promised to look into it…
Notice the important verb pattern to promise to do something, e.g. She promised to tidy her room but she obviously couldn’t be bothered.