Why are $e$ and $\pi$ believed to be normal? I've found that affirmation in several sources, but I can't think of an obvious reason. 
 A: I see two reasons:
1. Almost all real numbers are normal. In fact normal numbers are the rule, not the exception.
2. We know many digits and it appears that they are normal. See for example here. Of course, one cannot know that this trend will continue without a proof, but it is a piece of evidence.
A: $\pi$ and $e$ both have quite simple definitions. It is generally felt that something so simple ought to be either normal, or conspicuously and dramatically non-normal -- having a small subtle deviation from normality seems like something that ought to require a more involved definition. This is not any kind of proof, of course, but it would certainly be a surprise if the simple series for $\pi$ and $e$ could produce something as sneaky as an almost-but-not-quite normal number.
Since the numerical evidence is clear that these numbers are not "conspicuously and dramatically non-normal", being normal is the only remaining non-surprising guess.
A: We don't know whether $e+\pi$ is irrational, but I'll bet anything that it is because almost all numbers are irrational and there's no particular reason to believe $e+\pi$ is rational. Analogously, the following is a theorem:

Almost all real numbers are normal.

I do not know of any easy proof of this fact, but, for the interested reader, one can be found in this paper or in many of its references. Given that $e$ and $\pi$ arise more from properties useful in calculus and geometry than from any definition based on their digits, we'd expect their digits to act like other real numbers do - we defined them analytically, so why should their digits behave in a strange way? Especially given, as is noted in @Eff's answer, that their starting segments look like what we'd think a normal number should look like.
Perhaps we're wrong about this guess, but it would require deep and surprising connections to be drawn to convince us that $e$ or $\pi$ was so different from most other reals.
