Testing must be done in all the possible devices and browsers.
Automation testing provides you all the possible ways to test an Application in all the possible browsers.
You can go by manual or Automation ,this depends on your expertise.
Cross browsing testing helps you to understand different UI’s and their behaviours.
Try it by yourself. Take 3 browsers (Chrome, firefox and safari) and start testing your application and you can see the differences or we can say errors.
Intially, while following agile , sometimes its client requirement to do testing in all the browsers.
So, u can understand this if you follow agile and certain protocols.
And if not , u can suggest cross browser testing in your company too. It will surely helps you to understand better about testing scenarios and u can make proper test cases too.