TL;DR: It allows you to edit encrypted stuff without decrypting it.
For example, if you had a book of everyone's salaries, you can ask Bob to add $5000 to it. Normally you'd have to give him the key, but with homomorphic encryption you can just perform operations without decrypting it and having Bob know everyone's salaries.
This becomes really powerful for stuff that you need to work with lots of data without knowing the contents of - cloud computing, for example.
For example, if you had a book of everyone's salaries, you can ask Bob to add $5000 to it. Normally you'd have to give him the key, but with homomorphic encryption you can just perform operations without decrypting it and having Bob know everyone's salaries.
This becomes really powerful for stuff that you need to work with lots of data without knowing the contents of - cloud computing, for example.