The main use of the external ID field is to record a reference to some other system you have with this students record in it - you can enter here an ID from that system to refer to when looking up this student there.
Or you can just use it as a "master ID" for the student. As only you can edit this field the student cannot change it by changing their email address, name details, or anything else. You can put something meaning full to you that will help you identify them if needed (be that a reference number, their original email address and/or name details, whatever suits you usage).
Turn on display of External ID:
If you are using the "External ID" field then you will most likely want to turn on the display option show it shows in the "Student" and "Enrolment" screens etc.
Then you can see it (if there is an enteral ID entered for that student) in various places such as the Student and Enrolment screens etc.