Setting the database to the right compatibility level is very important as it can have a negative impact on the performance and behavior of your databases. I've seen that many customers upgrade their SQL Servers from 2008 -> 2012 or from 2012 -> 2014 but forget to upgrade the compatibility of their databases. This basically means that though you have an upgraded version of SQL Server, your database cannot benefit from any new or enhanced features of SQL Server since your database is running on an old compatibility level.
Incompatible database compatibility could also lead to inefficient query plans. Therefore, it is important that you upgrade the compatibility of databases right after the database is upgraded to a newer version of SQL Server. This script can help go thru all the databases and provide you with a report on what databases are on old compatibility level and what compatibility level they need to be upgraded to.