How to prevent people from reading what's on my phone when I'm on public transportation with them?

You can get yourself a privacy screen filter:

It works by severely reducing the viewing angle of the screen, so that it's only readable by the user who's right in front of it. For example, here's how it looks on an iPhone.

Unless you value privacy above anything else, I suggest you get a detachable model or a phone pocket with a privacy filter built into it. That way, you can still share something on your phone with people around you when you intend to.

In addition to the privacy screen suggested in the other answer, there's a couple of strategies I use for this:

• First and foremost, I wouldn't open or watch anything really private while on public transport. Just as you wouldn't read your credit card number out loud on the phone while in public you probably shouldn't read confidential e-mails on the train.
• I find that just tilting the phone a bit helps. Where I take public transport, I'm often standing, so there's not really much space to move around and just a slight tilt of the screen can make it really hard to read.
• I find it works when I read something in a different language, as people will lose interest quite quickly. This works very well if you are travelling or if happen to live in a country where your language is not very common. Of course, if your only language is English, this will not work most of the time.

One aspect that I feel is important (and which has been addressed in a couple of answers on IPS) is that the person might not be doing it on purpose. As I often have to stand on crammed trains, sometimes the most appropriate place to look at is somebody's screen. Or I might just be staring into space.

As a not so serious side note, I read this article about a new software switching to the camera and showing the person puking rainbows every time somebody peeks on your phone, that should work very well if it's every a real product.

For all those people who don't have a privacy screen filter and do not find the "solution" to put the phone away acceptable:

• Choose a window seat with seats where the orientation changes. The good
  seats are an A, the entrance is E, the bad seats are an F, explanation follows:

E |FA|AA|AF| E |FF|FF|FF|

Near the entrance it is very easy to see the phone from the window seat. The only exception is that there is an opaque screen behind you, in this case F changes to A, the seat is ok. Hold your phone so that the angle between window pane and screen is less than 45° and hold it also upright, this prevents reflections and being angled prevents that your direct neighbor can see it. Because the neighbor behind you sits normally with the head in the opposite direction, it would be very conspicous and rude if he tries to change his orientation on the seat to watch your screen (Exceptions are children, they are insatiably curious and often sit inverted, but in this case I normally don't mind if they are watching).

If you on the other hand are standing, you need a jacket. Put the phone against the inside of your jacket like if you are putting your wallet in and view it by glancing downwards and sideways. Disadvantage: It is a quite unnatural position, so the people who watched you see that you are hiding your phone and the people who did not watch you see only that you are hiding something in your jacket. So I recommend that only if an asshole does not stop trying to watch the screen despite being noticed by you and other people.