Conditional Logit

I was trying to understand how conditional logit and fixed effects in multinomial logit worked, to explain to someone who asked, and I failed. Greene's text was not very helpful. The best thing I found was some notes from Penn: "Conditional Logistic Regression (CLR) for Matched or Stratified Data". The bottom line seems to be that conditional logit (clogit in Stata) chooses its parameter estimates to maximize the likelihood of the variation we see within the strata, while ignoring variation across strata. Thus, if we have data on 30 people choosing to travel by either car or bus over 200 days, we could use 30 dummies for the people, but in conditional logit we don't. Also, in conditional logit, unlike logit with dummies, if someone always travels by car instead of varying, that person is useless to the estimation.