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To denote a point on this path we use a single coordinate that we call $\tau$. We think of an observer as tracing a path through spacetime. However, we can make sense of this statement. You can easily see that it does not make sense if you use ordinary definitions: the speed of light is measured in 'length per time', while a 'speed through time' would be measured by 'time per time', which is just a number. Now, the statement about traveling through time 'at the speed of light' needs to be qualified. The only complication could have been some numerical factor in front of $m$. If you accept that energy and mass are related, and you know that nature has a natural velocity $c$, then $E=mc^2$ is the simplest thing you can write that describes this. First, the fact that an object at rest has energy $mc^2$ is a simple matter of dimensional analysis.
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