The Event Log File viewer is a GUI wrapper around the Event Monitoring API. As such, it will only show you the details that are currently being reported by that API. By default you only get the login, logout, and insecure external asset events for free in a production org.
That said, the data returned from that API is not specific to the user accessing it.
If you want to get fine grained monitoring of API calls you will want the API event type, which is a paid add-on. Also, there isn't currently a single Event Type that will track all API usage. Please consider voting for the idea: Event Monitoring for all calls that contribute to the 24 hour API request limit
I've made notes previously about the various ways the the API usage can be monitored - Monitoring your Salesforce API usage. This includes the API Usage Last 7 Days report that Pranay pointed out, the limit headers that get added to REST and SOAP responses, and the dedicated REST limits resource.