A uniform distribution has a constant density over its support:
\(f(x)=\begin{cases}c&\text{for }a\le x\le b\\0&\text{otherwise}\end{cases}\)
Any probability distribution sums to 1 over its support, so in this case \(f(x)\) satisfies
\(\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=1\)
Replace \(f(x)\) with its definition as given above, so you have
\(1=\displaystyle\int_a^bc\,\mathrm dx=cx\bigg|_{x=a}^{x=b}=c(b-a)\)
\(\implies c=\dfrac1{b-a}\)
So the density function is
\(f(x)=\begin{cases}\dfrac1{b-a}&\text{for }a\le x\le b\\\\0&\text{otherwise}\end{cases}\)
