Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.

For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side of the $x$-axis), whereas a cursory glance would cause one to suspect that adding a positive amount should shift in the positive direction. Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand.

I generally explain this by saying $x$ is getting a "head start". For example, suppose $f(x)$ has a root at $x = 5$. The graph of $f(x+1)$ is getting a unit for free, and so we only need $x = 4$ to get the same output before as before (i.e. a root). Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left.

My explanation seems to help some students and mystify others. I was hoping someone else in the community had an enlightening way to explain these phenomena. Again, I emphasize that the purpose is to strengthen the student's intuition; a rigorous algebraic approach is not what I'm looking for.
Here is a setting where the students will have experience, having nothing to do with skills in algebra: daylight savings time (since you're in the USA).  When we advance clocks ahead by an hour we lose an hour of sleep.  Thus replacing $t$ with $t+1$ shifts your schedule for the day back by one hour. 

Here, $t$ is the old "function" and $t + 1$ is the new "function". $t$ is one hour behind $t + 1$, but after this transformation, $t + 1$ shortens the day by one hour. So in the end, we lose an hour.  Point out to the students that it also works the other way when we turn the clocks back ($t$ is replaced with $t-1$) and then gain an hour in our schedule.
Just as an extension to the KCd's answer, we can think of this not only as daylight saving problem, but also as a timezone problem. For example, if you leave New York for Singapore you have to shift time on your watch $+12$ hours in order to keep up with local Singapore time. But if you want to express Singapore time in terms of New York time (due to the jet lag, perhaps) you could say "it's $x - 12$ hours in Singapore now in terms of New York time" or "it's $x - 12$ hours in New York now".