Such a combination of letters is called a root.

So the root for salaam – islaam – muslim is 

s – l – m

س – ل – م

This particular root (s – l – m) conveys the idea of ‘peace’.  And this is what the three words actually mean:

salaam is the Arabic word for ‘peace’.

islaam is the name of the religion and means ‘submission’ (to the will of God/Allah), and this again means to be in peace with God/Allah and oneself. 

muslim means a person who has submitted their will to God/Allah and is, therefore, in peace with God/Allah and themselves.