y-intercept

Why it is the value at x = 0

The $y$-axis is the set of points where $x = 0$. Substituting $x = 0$ into $y = mx + b$:

$$y = m(0) + b = b$$

The $mx$ term disappears, and $b$ is what remains. That is the whole reason $b$ is the intercept — it is the only part of the equation that survives when $x$ is zero.

What it means in a real situation

The $y$-intercept is the value before anything changes: a flat fee before any usage, a balance before any deposits, a temperature at time zero. In $y = 15x + 25$ describing a gym membership, the $25 is owed before a single month passes.

The x-intercept is the mirror image

The $x$-intercept is where the graph crosses the horizontal axis, so $y = 0$ there. Set the other variable to zero and solve. Mixing the two up is the most common slip when graphing from intercepts — see graphing linear equations.