Hi, the word while is used to contrast. It’s used between two complete sentences.
For example
Dr. Hermit believes that antibiotic creams help reduce the spread of disease, while Dr. Samsa has a completely opposite opinion and feels that antibiotic creams are detrimental to people's immune systems in the long run.
In this case, while is used to contrast both doctors' different opinions.
The speaker is giving an explanation/background information of why they want to kill the old man to inform readers before they actually do it (mainly to tell readers their purpose/ thoughts of future actions).
