You can get some intuition for small changes in $X$ by considering
$$ \begin{align} Y+\delta Y&=\alpha + \beta \ln(X + \delta X)\\ &=\alpha + \beta \ln X + \beta\ln\left(1+\frac{\delta X}{X}\right)\\ \therefore\delta Y&\approx \beta\frac{\delta X}{X}, \end{align} $$$$ \begin{align} Y+\delta Y&=\alpha + \beta \ln(X + \delta X)\\ &=\underbrace{\alpha + \beta \ln X}_Y + \beta\ln\left(1+\frac{\delta X}{X}\right)\\ \therefore\delta Y&\approx \beta\frac{\delta X}{X}, \end{align} $$
where the approximation is obtained by a Taylor series expansion of the logarithm to first order in $\delta X$.
In other words, a 1% increase in $X$ gives you an increase of $0.01\beta$ in the dependent variable. But this only holds for small changes in $X$.