Manifest, as a verb, means to make something evident [obvious or easily recognized] or clear. It is used to describe when something is easily seen, understood, or revealed.
The teacher’s frustration manifested in her angry tone. (Her frustration was easily seen from the way she spoke.)
The success of the project manifested through the positive feedback from clients. (The success of the project was evident [obvious or easy to see] because clients gave positive feedback.)
Very Simple Definition:
Manifest means to show or make something clear. It is used when something becomes easy to see or understand.
The smile on her face manifested her happiness. (Her happiness was shown by the smile on her face.)