Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.
The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the definition of the probability mass function. I am not sure if my question makes sense, but how should I think of $ P( X = a) $ when $X$ is a continuous random variable?
Since probability, $\mathbb P$ is a measure, it is generally defined to mimic the notion of a distance in a given set but with extra conditions to capture reality. Intuitively, distance in $3$D is a volume, distance in $2$D is an area, distance in $1$D is length, distance in discrete/counting numbers is the value. If the notion of continuity existed in this setting, we can prove that infinitely cutting a sheet of paper into two to make a line would lead to something too infinitesimal compared to the sheet we started with. This notion is encoded in saying that the measure of a line in $2$D is zero. Similarly, the measure of a sheet in $3$D is zero and the measure of discrete/counting numbers in $1$D is zero. For the sake of intuition, I implicitly assumed that the pre-image of a discrete random variable is discrete, which in general, is not true. For instance, the indicator function of an interval on real line.