Why install logstash if I can just send the data through REST to elasticsearch?
No, you do not have to install Logstash, if you plan to collect, normalize and write your application data yourself. As you correctly assumed, Logstash would be a replacement for your PHP script.
Nevertheless, you might still consider to have a look at Logstash. Since it is developed and maintained by same company taking care of Elastic Search, you could benefit from upcoming changes and optimizations.
As you can read from the introduction, Logstash is a tool to read data from multiple sources, normalize it and write the result to multiple destinations. For more details on which sources, filters and oputputs Logstash offers, you should also take a look at the pipeline documentation.