Everything depends on your router model. The normal way to collect log entries from a router is via syslog.
There are two things you'll need to do:
Configure the router to send syslog data
Configure Splunk to receive the data.
For Cisco routers, the command is:
Of course, replace the IP address with that of your Splunk server. There are plenty of other syslog references out there. For Juniper, this may help. Many home-class routers have an option to export syslog as well, typically buried under an "Advanced" menu somewhere.