"They run because we run."
Translation:Ritheann siad mar rithimid.
You mean siad? One or two (mé and possibly muid, depending on dialect) of the verb forms are 'synthetic,' i.e. the subject pronoun is smooshed together with the verb. In the present tense, you have to smoosh when you say I go, I run, etc.--the -m on the end is what's left of mé (I) after smooshing). Most people smoosh with muid (we) and get rithimid--you can see that the -mid is what's left of muid.
So rithim means I run, and rithimid means we run. If you put mé or muid on those forms, you'd be saying I I run or We we run. All of the other verb forms are ritheann and just mean run, so you need to add a subject pronoun.
Check out all the forms of rith here: http://www.teanglann.ie/en/gram/rith
Is there a difference in emphasis between 'rithimid' and 'ritheann muid' or are they completely interchangeable? I am thinking of the distinction between 'corremos' and 'nosotros corremos' in Spanish where the inclusion of the pronoun can be used to emphasize that it is we who are running.