iOS 7, Facebook Messenger app keeps crashing
An app that consistently crashes on launch is usually due to the device being out of memory (RAM, not system storage). There is a nice list of ways to cure that here. 

The next likely cause is bad data triggering a bug. You can often delete the app and redownload it to clear bad data files. Keep in mind if the app downloads data from the internet or has a backup / restore function. 

Last resort is contacting the developer for specific recovery or triage steps based on their design of their app. Support links are provided in the App Store if you can’t find one searching the internet.