Thank you for reaching out to ask2.extension.

This is called lilac blight and the weather is responsible.  There is a copper based treatment, but that is applied in the spring.  Now all you can do is be sure to clean up the debris from the ground.  The blight can overwinter there.  
You can cut the bushes down and dispose of the infected parts, but that doesn’t mean it won’t come back.  If you decide to do that note that your lilacs will not bloom next spring.

I had one bush that had this and I did not treat it.  The second year even the flowers were affected.  I ended up pulling the bush.

If you decide to treat, be sure to read all of the instructions before using.